A lambda is written with an arrow as well as a call (#1151)

`a => a + 3` was a parse error and is now the same entity as
`lambda(a, a + 3)`. Several parameters are the curried form #495's plan
specifies: `a b => a + b` is `a => b => a + b`, which is
`lambda(a, b, a + b)`.

The nodes were never missing. `Lambda`, `Application`, beta reduction and
currying are all in the library already, and #1137's ODE solver is built on
them; what that plan describes and the library did not have is the syntax.
This is the half of it that costs nothing -- `=` followed by `>` was not a
token and not a parse, so no reading of any valid input changes. The other
half (`f a b`, `sin x`, `sin (x)`) changes what juxtaposition means, which is
the decision #286 is about, and is not free.

Written by sharing the left side rather than as `names+ '=>' body | expression`.
Both alternatives of that begin with a name and stay viable through a second
one, juxtaposition being multiplication, so `a b => a + b` was decided as a
product before the arrow was reached and came back "mismatched input '=>'".
Sharing leaves one decision, taken on the token after the left side, and the
parameters are read back out of the product it parsed as.

That also settles the parameter that cannot arrive as a name token: `i` lexes
as the imaginary unit, so `i => i + 1` had to be read the way `lambda(i, i + 1)`
is read -- through `Binding` -- and the two now agree. #976.

A parameter that is not a name is refused, as the plan says: `a 3 => 3`,
`2 => 3` and `x + 1 => 2` raise. They raised before too, under a different
name -- `UnhandledParseException` where they now raise
`InvalidArgumentParseException` -- which is recorded in BREAKING-CHANGES.

The arrow is read, not printed. A lambda still prints as `lambda(x, x + 1)`,
which is what keeps the round trip the printed form promises.

The grammar was regenerated the way ImproveParser.md sets out. The unmodified
grammar was regenerated first and its diff confirmed empty, so what is
committed here is the rule and not a toolchain version. Syntax.md updated in
three places.

Full suite 9386 passed, 0 failed.

Part of #495.


Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

Co-authored-by: Claude Opus 5 <noreply@anthropic.com>

Author: Rafael-SOWNet

BREAKING-CHANGES.md

@@ -106,6 +106,46 @@ read first.
 | | `Transformation.Factorization.Name` | `… then polynomial-factorization` | `… then polynomial-factorization then numeric-content` |
 | **Silent** | `"arccotan(-1)".ToEntity().Simplify()`, and every negative argument the inverse-trigonometric table knows | `3/4 * pi` — the textbook range, and **not equal to `arccotan(-1)`**, whose value is `-pi/4` | `-1/4 * pi` |
 | | `"e ^ ln(x)".ToEntity().Simplify()`, and every exponential of a natural logarithm | `e ^ ln(x)` — left as written | `x` |
+| | `"a => a + 3".ToEntity()`, and every lambda written with an arrow | `UnhandledParseException` | `lambda(a, a + 3)` |
+
+### A lambda is written with an arrow as well as a call
+
+`a => a + 3` was a parse error and is now the same entity as `lambda(a, a + 3)`. Several
+parameters are the curried form the plan in
+[#495](https://github.com/asc-community/AngouriMath/issues/495) specifies — `a b => a + b` is
+`a => b => a + b`, which is `lambda(a, b, a + b)`.
+
+```
+"a => a + 3".ToEntity()                          was  UnhandledParseException
+                                                 is   lambda(a, a + 3)
+"a b => a + b".ToEntity()                        is   lambda(a, lambda(b, a + b))
+"apply(apply(a b => a + b, 1), 2)".Simplify()    is   3
+```
+
+**Nothing that parsed before parses differently.** `=` followed by `>` was not a token and not a
+parse, so no reading of any valid input has changed; `>=`, `->`, `=` and the rest are untouched,
+and there are tests pinning them. The `Lambda` node, beta reduction and currying were all already
+there — this is the syntax for them.
+
+**The arrow is read, not printed.** A lambda still prints as `lambda(x, x + 1)`, which is what
+keeps the round trip the printed form promises: several spellings may be read, exactly one is
+printed.
+
+**Every parameter must be a name**, which is what the plan says. `a 3 => 3` is refused, and so
+are `2 => 3` and `x + 1 => 2`. Those raised `UnhandledParseException` before and now raise
+`InvalidArgumentParseException` — still invalid, differently named. Code catching the parse
+exception by type around input like that will not catch this one.
+
+An index called `i` is the name rather than the imaginary unit, matching `lambda(i, i + 1)`,
+which it gets by reading its parameters through the same `Binding` the call form uses
+([#976](https://github.com/asc-community/AngouriMath/issues/976)).
+
+**Not in this change:** the rest of that plan's syntax — `f a b` for `apply(apply(f, a), b)`,
+`sin x` without brackets, and `sin (x)` with a space. Each of those changes what juxtaposition
+means, which is the decision [#286](https://github.com/asc-community/AngouriMath/issues/286) is
+about, and none of them is free the way the arrow is.
+
+[#495](https://github.com/asc-community/AngouriMath/issues/495).
 
 ### The exponential of a natural logarithm folds
 

Sources/.editorconfig

@@ -171,5 +171,8 @@ file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed
 [Tests/UnitTests/Algebra/SolveTest/ParametricSystemTest.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 
+[Tests/UnitTests/Common/LambdaArrowSyntaxTest.cs]
+file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
+
 [Tests/UnitTests/Calculus/ApplicationDerivativeTest.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n

Sources/AngouriMath/Core/Antlr/AngouriMath.g

@@ -243,8 +243,49 @@ Nodes end
 
 */
 
+/* `a => a + 3`, and `a b => a + b` for `a => b => a + b`. Lowest priority of anything, and
+   right-associative through the recursion on `expression`, so the body runs to the end of what
+   is being parsed: `a => a + 3` is `a => (a + 3)` and never `(a => a) + 3`.
+
+   The parameters are VARIABLE tokens rather than expressions, which is what makes `a 3 => 3`
+   invalid as the plan says it should be: `3` cannot match, the alternative fails, and what is
+   left is not a parse.
+
+   The body is built exactly as `lambda(...)` builds it, through Binding, so that the two spell
+   the same thing -- including the case a lambda's Variable-typed parameter cannot state
+   directly, an index called `i`. https://github.com/asc-community/AngouriMath/issues/976 */
 expression returns[Entity value]
     : s = provided_expression { $value = $s.value; }
+      ('=>' b = expression
+        {
+            /* The parameters are read back out of an ordinary expression rather than matched as
+               a list of names, and that is about what the parser can predict rather than about
+               taste. Written as `names+ '=>' body | expression`, both alternatives begin with a
+               name and stay viable through a second one -- juxtaposition being multiplication --
+               so `a b => a + b` was decided as a product before the arrow was ever reached, and
+               came back "mismatched input '=>'". Sharing the left side leaves one decision, taken
+               on the token after it.
+
+               So `a b c` arrives here as the product it parsed as, and its factors in order are
+               the parameters. Anything that is not a name fails here instead of failing to
+               parse: `a 3 => 3`, which the plan calls invalid, raises rather than being read as
+               a lambda over `a` and `3`.
+
+               Through Binding, exactly as `lambda(...)` builds it, so the two spell the same
+               thing -- including an index called `i`, which lexes as the imaginary unit and can
+               therefore never arrive as a Variable token at all.
+               https://github.com/asc-community/AngouriMath/issues/976 */
+            Entity lambdaBody = $b.value;
+            var parameters = ($value is Mulf ? Mulf.LinearChildren($value) : new[] { $value }).ToList();
+            foreach (var x in ((IEnumerable<Entity>)parameters).Reverse())
+            {
+                var bound = AngouriMath.Core.Binding.Of(x);
+                if (bound.Name is not Variable v) throw new InvalidArgumentParseException($"Lambda is expected to have valid parameters, {x} encountered instead");
+                lambdaBody = bound.In(lambdaBody).LambdaOver(v);
+            }
+            $value = lambdaBody;
+        }
+      )?
     ;
 
 

Sources/AngouriMath/Core/Antlr/AngouriMath.interp

@@ -143,14 +143,15 @@ T__141=142
 T__142=143
 T__143=144
 T__144=145
-NEWLINE=146
-NUMBER=147
-SPECIALSET=148
-BOOLEAN=149
-NAN=150
-VARIABLE=151
-COMMENT=152
-WS=153
+T__145=146
+NEWLINE=147
+NUMBER=148
+SPECIALSET=149
+BOOLEAN=150
+NAN=151
+VARIABLE=152
+COMMENT=153
+WS=154
 '!'=1
 '^'=2
 '-'=3
@@ -180,120 +181,121 @@ WS=153
 'implies'=27
 '->'=28
 'provided'=29
-','=30
-';'=31
-':'=32
-'+oo'=33
-'-oo'=34
-'(|'=35
-'|)'=36
-'['=37
-']T'=38
-']'=39
-'('=40
-')'=41
-'{'=42
-'}'=43
-'log('=44
-'log10('=45
-'log2('=46
-'pow('=47
-'sqrt('=48
-'cbrt('=49
-'sqr('=50
-'ln('=51
-'exp('=52
-'sin('=53
-'cos('=54
-'tan('=55
-'cotan('=56
-'cot('=57
-'sec('=58
-'cosec('=59
-'csc('=60
-'arcsin('=61
-'arccos('=62
-'arctan('=63
-'arccotan('=64
-'arcsec('=65
-'arccosec('=66
-'arccsc('=67
-'acsc('=68
-'asin('=69
-'acos('=70
-'atan('=71
-'acotan('=72
-'asec('=73
-'acosec('=74
-'acot('=75
-'arccot('=76
-'sinh('=77
-'sh('=78
-'cosh('=79
-'ch('=80
-'tanh('=81
-'th('=82
-'cotanh('=83
-'coth('=84
-'cth('=85
-'sech('=86
-'sch('=87
-'cosech('=88
-'csch('=89
-'asinh('=90
-'arsinh('=91
-'arsh('=92
-'arcsinh('=93
-'acosh('=94
-'arcosh('=95
-'arch('=96
-'arccosh('=97
-'atanh('=98
-'artanh('=99
-'arth('=100
-'arctanh('=101
-'acoth('=102
-'arcoth('=103
-'acotanh('=104
-'arcotanh('=105
-'arcth('=106
-'arccotanh('=107
-'asech('=108
-'arsech('=109
-'arsch('=110
-'arcsech('=111
-'acosech('=112
-'arcosech('=113
-'arcsch('=114
-'arccosech('=115
-'acsch('=116
-'factorial('=117
-'gamma('=118
-'derivative('=119
-'integral('=120
-'limit('=121
-'limitleft('=122
-'limitright('=123
-'sum('=124
-'product('=125
-'signum('=126
-'sgn('=127
-'sign('=128
-'abs('=129
-'phi('=130
-'floor('=131
-'ceil('=132
-'ceiling('=133
-'round('=134
-'min('=135
-'max('=136
-'gcd('=137
-'trunc('=138
-'lcm('=139
-'erf('=140
-'conjugate('=141
-'domain('=142
-'piecewise('=143
-'apply('=144
-'lambda('=145
-'NaN'=150
+'=>'=30
+','=31
+';'=32
+':'=33
+'+oo'=34
+'-oo'=35
+'(|'=36
+'|)'=37
+'['=38
+']T'=39
+']'=40
+'('=41
+')'=42
+'{'=43
+'}'=44
+'log('=45
+'log10('=46
+'log2('=47
+'pow('=48
+'sqrt('=49
+'cbrt('=50
+'sqr('=51
+'ln('=52
+'exp('=53
+'sin('=54
+'cos('=55
+'tan('=56
+'cotan('=57
+'cot('=58
+'sec('=59
+'cosec('=60
+'csc('=61
+'arcsin('=62
+'arccos('=63
+'arctan('=64
+'arccotan('=65
+'arcsec('=66
+'arccosec('=67
+'arccsc('=68
+'acsc('=69
+'asin('=70
+'acos('=71
+'atan('=72
+'acotan('=73
+'asec('=74
+'acosec('=75
+'acot('=76
+'arccot('=77
+'sinh('=78
+'sh('=79
+'cosh('=80
+'ch('=81
+'tanh('=82
+'th('=83
+'cotanh('=84
+'coth('=85
+'cth('=86
+'sech('=87
+'sch('=88
+'cosech('=89
+'csch('=90
+'asinh('=91
+'arsinh('=92
+'arsh('=93
+'arcsinh('=94
+'acosh('=95
+'arcosh('=96
+'arch('=97
+'arccosh('=98
+'atanh('=99
+'artanh('=100
+'arth('=101
+'arctanh('=102
+'acoth('=103
+'arcoth('=104
+'acotanh('=105
+'arcotanh('=106
+'arcth('=107
+'arccotanh('=108
+'asech('=109
+'arsech('=110
+'arsch('=111
+'arcsech('=112
+'acosech('=113
+'arcosech('=114
+'arcsch('=115
+'arccosech('=116
+'acsch('=117
+'factorial('=118
+'gamma('=119
+'derivative('=120
+'integral('=121
+'limit('=122
+'limitleft('=123
+'limitright('=124
+'sum('=125
+'product('=126
+'signum('=127
+'sgn('=128
+'sign('=129
+'abs('=130
+'phi('=131
+'floor('=132
+'ceil('=133
+'ceiling('=134
+'round('=135
+'min('=136
+'max('=137
+'gcd('=138
+'trunc('=139
+'lcm('=140
+'erf('=141
+'conjugate('=142
+'domain('=143
+'piecewise('=144
+'apply('=145
+'lambda('=146
+'NaN'=151