From eb8c9313694e56912780f907399cf56e7d5bc75c Mon Sep 17 00:00:00 2001
From: Sergei Winitzki <winitzki@gmail.com>
Date: Tue, 3 Apr 2018 22:38:38 -0700
Subject: [PATCH 01/11] implement MatchE simplification of nested match

---
 src/main/scala/io/chymyst/ch/TermExpr.scala | 32 +++++++++++++++------
 1 file changed, 24 insertions(+), 8 deletions(-)

diff --git a/src/main/scala/io/chymyst/ch/TermExpr.scala b/src/main/scala/io/chymyst/ch/TermExpr.scala
index 6829ca4..2e92fab 100644
--- a/src/main/scala/io/chymyst/ch/TermExpr.scala
+++ b/src/main/scala/io/chymyst/ch/TermExpr.scala
@@ -803,36 +803,52 @@ final case class MatchE(term: TermExpr, cases: List[TermExpr]) extends TermExpr
   }
 
   private[ch] override def simplifyOnceInternal(withEta: Boolean): TermExpr = {
-    lazy val casesSimplified = cases.map(_.simplifyOnce(withEta))
+    val ncases = cases.length
     term.simplifyOnce(withEta) match {
       // Match a fixed part of the disjunction; can be simplified to just one clause.
       // Example: Left(a) match { case Left(x) => f(x); case Right(y) => ... } can be simplified to just f(a).
       case DisjunctE(index, total, termInjected, _) ⇒
-        if (total === cases.length) {
+        if (total === ncases) {
           AppE(cases(index).simplifyOnce(withEta), termInjected).simplifyOnce(withEta)
-        } else throw new Exception(s"Internal error: MatchE with ${cases.length} cases applied to DisjunctE with $total parts, but must be of equal size")
+        } else throw new Exception(s"Internal error: MatchE with $ncases cases applied to DisjunctE with $total parts, but must be of equal size")
 
       // Match of an inner match, can be simplified to a single match.
-      // Example: (Left(a) match { case Left(x) ⇒ ...; case Right(y) ⇒ ... }) match { case ... ⇒ ... }
-      // can be simplified to Left(a) match { case Left(x) ⇒ ... match { case ... ⇒ ...}; case Right(y) ⇒ ... match { case ... ⇒ ... } }
+      // Example: (q match { case Left(x) ⇒ ...; case Right(y) ⇒ ... }) match { case ... ⇒ ... }
+      // can be simplified to q match { case Left(x) ⇒ ... match { case ... ⇒ ...}; case Right(y) ⇒ ... match { case ... ⇒ ... } }
       case MatchE(innerTerm, innerCases) ⇒
         MatchE(innerTerm, innerCases map { case CurriedE(List(head), body) ⇒ CurriedE(List(head), MatchE(body, cases)) })
+          .simplifyOnce(withEta)
 
       // Detect the identity patterns:
       // MatchE(_, List(a ⇒ DisjunctE(0, total, a, _), a ⇒ DisjunctE(1, total, a, _), ...))
       // MatchE(_, a: T1 ⇒ DisjunctE(i, total, NamedConjunctE(List(ProjectE(0, a), Project(1, a), ...), T1), ...), _)
       case termSimplified ⇒
-        if (cases.nonEmpty && {
+
+        // Replace redundant matches on the same term, can be simplified by eliminating one match subexpresssion.
+        // Example: q match { case x ⇒ q match { case y ⇒ b; case other ⇒ ... } ... }
+        // We already know that q was matched as Left(x). Therefore, we can replace y by x in b and remove the `case other` clause altogether.
+
+        val casesSimplified = cases.zipWithIndex.map { case (c@CurriedE(List(headVar), _), i) ⇒
+          TermExpr.substMap(c.simplifyOnce(withEta)) {
+            case MatchE(otherTerm, otherCases) if otherTerm === termSimplified ⇒
+              // We already matched `otherTerm`, and we are now in case `c`, which is `case x ⇒ ...`.
+              // Therefore we can discard any of the `otherCases` except the one corresponding to `c`.
+              // We can replace the `q match { case y ⇒ b; ...}` by `b` after replacing `x` by `y` in `b`.
+              val remainingCase = otherCases(i)
+              AppE(remainingCase, headVar).simplifyOnce(withEta)
+          }
+        }
+        if (casesSimplified.nonEmpty && {
           casesSimplified.zipWithIndex.forall {
             // Detect a ⇒ a pattern
             case (CurriedE(List(head@VarE(_, _)), body@VarE(_, _)), _)
               if head.name === body.name
             ⇒ true
             case (CurriedE(List(head@VarE(_, _)), DisjunctE(i, len, x, _)), ind)
-              if x === head && len === cases.length && ind === i
+              if x === head && len === ncases && ind === i
             ⇒ true
             case (CurriedE(List(head@VarE(_, headT)), DisjunctE(i, len, NamedConjunctE(projectionTerms, conjT), _)), ind) ⇒
-              len === cases.length && ind === i && headT === conjT &&
+              len === ncases && ind === i && headT === conjT &&
                 projectionTerms.zipWithIndex.forall {
                   case (ProjectE(k, head1), j) if k === j && head1 === head ⇒ true
                   case _ ⇒ false

From a43f1531642936538e2291465d3b1ea69e3c22ad Mon Sep 17 00:00:00 2001
From: Sergei Winitzki <winitzki@gmail.com>
Date: Tue, 3 Apr 2018 23:08:58 -0700
Subject: [PATCH 02/11] reintroduce simplify/distinct into theorem prover

---
 src/main/scala/io/chymyst/ch/TermExpr.scala      | 4 +++-
 src/main/scala/io/chymyst/ch/TheoremProver.scala | 3 ++-
 2 files changed, 5 insertions(+), 2 deletions(-)

diff --git a/src/main/scala/io/chymyst/ch/TermExpr.scala b/src/main/scala/io/chymyst/ch/TermExpr.scala
index 2e92fab..8e59530 100644
--- a/src/main/scala/io/chymyst/ch/TermExpr.scala
+++ b/src/main/scala/io/chymyst/ch/TermExpr.scala
@@ -835,7 +835,9 @@ final case class MatchE(term: TermExpr, cases: List[TermExpr]) extends TermExpr
               // Therefore we can discard any of the `otherCases` except the one corresponding to `c`.
               // We can replace the `q match { case y ⇒ b; ...}` by `b` after replacing `x` by `y` in `b`.
               val remainingCase = otherCases(i)
-              AppE(remainingCase, headVar).simplifyOnce(withEta)
+              val result = AppE(remainingCase, headVar).simplifyOnce(withEta)
+              println(s"DEBUG: replacing ${MatchE(otherTerm, otherCases)} by $result in ${c.simplifyOnce(withEta)}")
+              result
           }
         }
         if (casesSimplified.nonEmpty && {
diff --git a/src/main/scala/io/chymyst/ch/TheoremProver.scala b/src/main/scala/io/chymyst/ch/TheoremProver.scala
index 6d7e53a..5317f20 100644
--- a/src/main/scala/io/chymyst/ch/TheoremProver.scala
+++ b/src/main/scala/io/chymyst/ch/TheoremProver.scala
@@ -93,9 +93,10 @@ object TheoremProver {
       val transformedProofs = explodedNewProofs.map(ruleResult.backTransform)
       val t1 = System.currentTimeMillis()
 
-      val result = transformedProofs.sortBy(_.informationLossScore).take(maxTermsToSelect(sequent))
+      val result = transformedProofs.map(_.simplifyOnce(withEta = false)).distinct.sortBy(_.informationLossScore).take(maxTermsToSelect(sequent))
       // Note: at this point, it is a mistake to do prettyRename, because we are calling this function recursively.
       // We will call prettyRename() at the very end of the proof search.
+      // It is also a mistake to do a `.simplifyOnce(withEta = true)`. The eta-conversion produces incorrect code here.
       if (debug) {
         println(s"DEBUG: elapsed ${System.currentTimeMillis() - t0} ms, .map(_.simplify()).distinct took ${System.currentTimeMillis() - t1} ms, produced ${result.size} terms out of ${transformedProofs.size} back-transformed terms; after rule ${ruleResult.ruleName} for sequent $sequent")
         //        println(s"DEBUG: for sequent $sequent, after rule ${ruleResult.ruleName}, transformed ${transformedProofs.length} proof terms:\n ${transformedProofs.mkString(" ;\n ")} ,\nafter simplifying:\n ${result.mkString(" ;\n ")} .")

From fa2cd8c89711b71358913e8375f1c4705c730e81 Mon Sep 17 00:00:00 2001
From: Sergei Winitzki <winitzki@gmail.com>
Date: Tue, 3 Apr 2018 23:12:19 -0700
Subject: [PATCH 03/11] bump version

---
 build.sbt | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/build.sbt b/build.sbt
index 424c652..4dd65ac 100644
--- a/build.sbt
+++ b/build.sbt
@@ -103,7 +103,7 @@ lazy val curryhoward: Project = (project in file("."))
   .settings(common)
   .settings(
     organization := "io.chymyst",
-    version := "0.3.7",
+    version := "0.3.8",
 
     licenses := Seq("Apache License, Version 2.0" -> url("https://www.apache.org/licenses/LICENSE-2.0.txt")),
     homepage := Some(url("https://github.com/Chymyst/curryhoward")),

From 820a4b97dc76e2b9ddd67c2b0c75f5af9721fdfa Mon Sep 17 00:00:00 2001
From: Sergei Winitzki <winitzki@gmail.com>
Date: Wed, 4 Apr 2018 00:33:28 -0700
Subject: [PATCH 04/11] generate variables for disjunctions

---
 docs/Tutorial.md                              |  4 +--
 src/main/scala/io/chymyst/ch/TermExpr.scala   | 12 +++++++
 src/main/tut/Tutorial.md                      |  4 +--
 .../io/chymyst/ch/unit/TermExprSpec.scala     | 31 +++++++++++++++----
 4 files changed, 41 insertions(+), 10 deletions(-)

diff --git a/docs/Tutorial.md b/docs/Tutorial.md
index 34706e2..4d28e93 100644
--- a/docs/Tutorial.md
+++ b/docs/Tutorial.md
@@ -888,8 +888,8 @@ res26: Boolean = true
 | `a.substTypeVar(b, c)` | `TermExpr ⇒ (TermExpr, TermExpr) ⇒ TermExpr` | replace a type variable in `a`; the type variable is specified as the type of `b`, and the replacement type is specified as the type of `c` |
 | `a.substTypeVars(s)` | `TermExpr ⇒ Map[TP, TypeExpr] ⇒ TermExpr` | replace all type variables in `a` according to the given substitution map `s` -- all type variables are substituted at once |
 | `u()`  | `TermExpr ⇒ () ⇒ TermExpr` and `TypeExpr ⇒ () ⇒ TermExpr` | create a "named Unit" term of type `u.t` -- the type of `u` must be a named unit type, e.g. `None.type` or a case class with no constructors |
-| `c(x...)`  | `TermExpr ⇒ TermExpr* ⇒ TermExpr` and `TypeExpr ⇒ TermExpr* ⇒ TermExpr` | create a named conjunction term of type `c.t` -- the type of `c` must be a conjunction whose parts match the types of the arguments `x...` |
-| `d(x)`  |  `TermExpr ⇒ TermExpr ⇒ TermExpr` and `TypeExpr ⇒ TermExpr ⇒ TermExpr` | create a disjunction term of type `d.t` using term `x` -- the type of `x` must match one of the disjunction parts in the type `d`, which must be a disjunction type |
+| `c(x...)`  | `TypeExpr ⇒ TermExpr* ⇒ TermExpr` and `TypeExpr ⇒ TermExpr* ⇒ TermExpr` | create a named conjunction term of type `c` -- the type `c` must be a conjunction whose parts match the types of the arguments `x...` |
+| `d(x)`  |  `TypeExpr ⇒ TermExpr ⇒ TermExpr` and `TypeExpr ⇒ TermExpr ⇒ TermExpr` | create a disjunction term of type `d` using term `x` -- the type of `x` must match one of the disjunction parts in the type `d`, which must be a disjunction type |
 | `c(i)` | `TermExpr ⇒ Int ⇒ TermExpr` | project a conjunction term onto part with given zero-based index -- the type of `c` must be a conjunction with sufficiently many parts |
 | `c("id")` | `TermExpr ⇒ String ⇒ TermExpr` | project a conjunction term onto part with given accessor name -- the type of `c` must be a named conjunction that supports this accessor |
 | `d.cases(x =>: ..., y =>: ..., ...)` | `TermExpr ⇒ TermExpr* ⇒ TermExpr` | create a term that pattern-matches on the given disjunction term -- the type of `d` must be a disjunction whose arguments match the arguments `x`, `y`, ... of the given case clauses |
diff --git a/src/main/scala/io/chymyst/ch/TermExpr.scala b/src/main/scala/io/chymyst/ch/TermExpr.scala
index 8e59530..85f024e 100644
--- a/src/main/scala/io/chymyst/ch/TermExpr.scala
+++ b/src/main/scala/io/chymyst/ch/TermExpr.scala
@@ -342,6 +342,18 @@ object TermExpr {
   }
 
   private[ch] def roundFactor(x: Double): Int = math.round(x * 10000).toInt
+
+  /** Generate all necessary free variables for equality checking of functions that consume disjunction types.
+    *
+    * @param typeExpr The type of the argument expression.
+    * @return A sequence of [[TermExpr]] values containing the necessary free variables.
+    */
+  def subtypeVars(typeExpr: TypeExpr): Seq[TermExpr] = typeExpr match {
+    case dt@DisjunctT(constructor, typeParams, terms) ⇒ terms.zipWithIndex.flatMap { case (t, i) ⇒ subtypeVars(t).map(v ⇒ DisjunctE(i, terms.length, v, dt)) }
+    case nct@NamedConjunctT(constructor, typeParams, accessors, wrapped) ⇒
+      TheoremProver.explode(wrapped.map(subtypeVars)).map(NamedConjunctE(_, nct))
+    case _ ⇒ Seq(VarE(freshIdents(), typeExpr))
+  }
 }
 
 sealed trait TermExpr {
diff --git a/src/main/tut/Tutorial.md b/src/main/tut/Tutorial.md
index 9185776..1a37837 100644
--- a/src/main/tut/Tutorial.md
+++ b/src/main/tut/Tutorial.md
@@ -725,8 +725,8 @@ getIdAutoTerm equiv getId.prettyRename
 | `a.substTypeVar(b, c)` | `TermExpr ⇒ (TermExpr, TermExpr) ⇒ TermExpr` | replace a type variable in `a`; the type variable is specified as the type of `b`, and the replacement type is specified as the type of `c` |
 | `a.substTypeVars(s)` | `TermExpr ⇒ Map[TP, TypeExpr] ⇒ TermExpr` | replace all type variables in `a` according to the given substitution map `s` -- all type variables are substituted at once |
 | `u()`  | `TermExpr ⇒ () ⇒ TermExpr` and `TypeExpr ⇒ () ⇒ TermExpr` | create a "named Unit" term of type `u.t` -- the type of `u` must be a named unit type, e.g. `None.type` or a case class with no constructors |
-| `c(x...)`  | `TermExpr ⇒ TermExpr* ⇒ TermExpr` and `TypeExpr ⇒ TermExpr* ⇒ TermExpr` | create a named conjunction term of type `c.t` -- the type of `c` must be a conjunction whose parts match the types of the arguments `x...` |
-| `d(x)`  |  `TermExpr ⇒ TermExpr ⇒ TermExpr` and `TypeExpr ⇒ TermExpr ⇒ TermExpr` | create a disjunction term of type `d.t` using term `x` -- the type of `x` must match one of the disjunction parts in the type `d`, which must be a disjunction type |
+| `c(x...)`  | `TypeExpr ⇒ TermExpr* ⇒ TermExpr` and `TypeExpr ⇒ TermExpr* ⇒ TermExpr` | create a named conjunction term of type `c` -- the type `c` must be a conjunction whose parts match the types of the arguments `x...` |
+| `d(x)`  |  `TypeExpr ⇒ TermExpr ⇒ TermExpr` and `TypeExpr ⇒ TermExpr ⇒ TermExpr` | create a disjunction term of type `d` using term `x` -- the type of `x` must match one of the disjunction parts in the type `d`, which must be a disjunction type |
 | `c(i)` | `TermExpr ⇒ Int ⇒ TermExpr` | project a conjunction term onto part with given zero-based index -- the type of `c` must be a conjunction with sufficiently many parts |
 | `c("id")` | `TermExpr ⇒ String ⇒ TermExpr` | project a conjunction term onto part with given accessor name -- the type of `c` must be a named conjunction that supports this accessor |
 | `d.cases(x =>: ..., y =>: ..., ...)` | `TermExpr ⇒ TermExpr* ⇒ TermExpr` | create a term that pattern-matches on the given disjunction term -- the type of `d` must be a disjunction whose arguments match the arguments `x`, `y`, ... of the given case clauses |
diff --git a/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala b/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala
index f3239a1..33fe584 100644
--- a/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala
+++ b/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala
@@ -15,8 +15,29 @@ class TermExprSpec extends FlatSpec with Matchers {
 
   behavior of "TermExpr miscellaneous methods"
 
+  it should "generate variables for disjunction subtypes" in {
+    val p = freshVar[Either[Option[Option[(Int, Int)]], Option[(Option[Int], Option[Int])]]]
+
+    val subtypeVars = TermExpr.subtypeVars(p.t).map(_.prettyPrint)
+
+    val indices = "z([0-9]+)".r.findAllMatchIn(subtypeVars.mkString("")).map(_.group(1).toInt).toList
+    indices.map(_ - indices.min) shouldEqual Seq(0, 1, 3, 2, 2, 3)
+
+    subtypeVars.map(_.replaceAll("z[0-9]+", "z")) shouldEqual List(
+      "(Left((None() + 0)) + 0)",
+      "(Left((0 + Some((None() + 0)))) + 0)",
+      "(Left((0 + Some((0 + Some(Tuple2(z, z)))))) + 0)",
+      "(0 + Right((None() + 0)))", "(0 + Right((0 + Some(Tuple2((None() + 0), (None() + 0))))))",
+      "(0 + Right((0 + Some(Tuple2((None() + 0), (0 + Some(z)))))))",
+      "(0 + Right((0 + Some(Tuple2((0 + Some(z)), (None() + 0))))))",
+      "(0 + Right((0 + Some(Tuple2((0 + Some(z)), (0 + Some(z)))))))"
+    )
+
+  }
+
   it should "compute identity function" in {
     def idAB[A, B] = TermExpr.id(typeExpr[A ⇒ B])
+
     idAB.prettyPrint shouldEqual "x ⇒ x"
     idAB.toString shouldEqual "\\((x:A ⇒ B) ⇒ x)"
     idAB.t.prettyPrint shouldEqual "(A ⇒ B) ⇒ A ⇒ B"
@@ -274,9 +295,9 @@ a ⇒ Tuple2(a._2._2, a._2._2) // Choose second element of second inner tuple.
 
     //    println(flattens.size)
 
-    def f[A] = allOfType[Option[(A, A, A)] ⇒ Option[(A, A, A)]]()
+    def f[A] = anyOfType[Option[(A, A, A)] ⇒ Option[(A, A, A)]]()
 
-    println(f.size)
+    f.size shouldEqual 28
     //    f[Int].map(_.lambdaTerm.prettyPrint).sorted.foreach(println)
     //    f.size shouldEqual factorial(4)
   }
@@ -284,9 +305,7 @@ a ⇒ Tuple2(a._2._2, a._2._2) // Choose second element of second inner tuple.
   it should "generate match clauses" in {
     def f[A] = anyOfType[Option[Option[A]] ⇒ Option[Option[A]]]().map(_.lambdaTerm)
 
-    println(f.size)
-    f.map(_.prettyPrint).foreach(println)
-
+    f.size shouldEqual 13
+    //    f.map(_.prettyPrint).foreach(println)
   }
-
 }

From 72670af9806de8ae329bf21bc7a2c304aa3ad3be Mon Sep 17 00:00:00 2001
From: Sergei Winitzki <winitzki@gmail.com>
Date: Wed, 4 Apr 2018 01:02:56 -0700
Subject: [PATCH 05/11] implement subtype fresh vars checking

---
 src/main/scala/io/chymyst/ch/TermExpr.scala   | 20 +++++++++++++++++--
 ...ecking.scala => SymbolicLawChecking.scala} |  6 +++---
 .../scala/io/chymyst/ch/unit/LJTSpec4.scala   |  2 +-
 .../io/chymyst/ch/unit/LambdaTermsSpec.scala  |  8 ++++----
 .../io/chymyst/ch/unit/TermExprSpec.scala     |  3 +++
 5 files changed, 29 insertions(+), 10 deletions(-)
 rename src/main/scala/io/chymyst/ch/data/{LawChecking.scala => SymbolicLawChecking.scala} (89%)

diff --git a/src/main/scala/io/chymyst/ch/TermExpr.scala b/src/main/scala/io/chymyst/ch/TermExpr.scala
index 85f024e..72adfe2 100644
--- a/src/main/scala/io/chymyst/ch/TermExpr.scala
+++ b/src/main/scala/io/chymyst/ch/TermExpr.scala
@@ -343,10 +343,10 @@ object TermExpr {
 
   private[ch] def roundFactor(x: Double): Int = math.round(x * 10000).toInt
 
-  /** Generate all necessary free variables for equality checking of functions that consume disjunction types.
+  /** Generate all necessary fresh variables for equality checking of functions that consume disjunction types.
     *
     * @param typeExpr The type of the argument expression.
-    * @return A sequence of [[TermExpr]] values containing the necessary free variables.
+    * @return A sequence of [[TermExpr]] values containing the necessary fresh variables.
     */
   def subtypeVars(typeExpr: TypeExpr): Seq[TermExpr] = typeExpr match {
     case dt@DisjunctT(constructor, typeParams, terms) ⇒ terms.zipWithIndex.flatMap { case (t, i) ⇒ subtypeVars(t).map(v ⇒ DisjunctE(i, terms.length, v, dt)) }
@@ -354,6 +354,22 @@ object TermExpr {
       TheoremProver.explode(wrapped.map(subtypeVars)).map(NamedConjunctE(_, nct))
     case _ ⇒ Seq(VarE(freshIdents(), typeExpr))
   }
+
+  /** Extensional equality check. If the term expressions are functions, fresh variables are substituted as arguments and the results are compared with `equiv`.
+    *
+    * @param termExpr1 The first term.
+    * @param termExpr2 The second term.
+    * @return `true` if the terms are extensionally equal.
+    */
+  def extEquals(termExpr1: TermExpr, termExpr2: TermExpr): Boolean = {
+    val t1 = termExpr1.simplify
+    val t2 = termExpr2.simplify
+    (t1.t === t2.t) && ((t1 equiv t2) || ((t1, t2) match {
+      case (CurriedE(h1 :: _, _), CurriedE(_ :: _, _)) ⇒
+        subtypeVars(h1.t).forall(term ⇒ extEquals(t1(term), t2(term)))
+      case _ ⇒ false
+    }))
+  }
 }
 
 sealed trait TermExpr {
diff --git a/src/main/scala/io/chymyst/ch/data/LawChecking.scala b/src/main/scala/io/chymyst/ch/data/SymbolicLawChecking.scala
similarity index 89%
rename from src/main/scala/io/chymyst/ch/data/LawChecking.scala
rename to src/main/scala/io/chymyst/ch/data/SymbolicLawChecking.scala
index 18fc221..f1a655b 100644
--- a/src/main/scala/io/chymyst/ch/data/LawChecking.scala
+++ b/src/main/scala/io/chymyst/ch/data/SymbolicLawChecking.scala
@@ -2,14 +2,14 @@ package io.chymyst.ch.data
 
 import io.chymyst.ch._
 
-object LawChecking {
+object SymbolicLawChecking {
 
   def checkFlattenAssociativity(fmap: TermExpr, flatten: TermExpr): Boolean = {
     // fmap ftn . ftn = ftn . ftn
     val lhs = flatten :@@ flatten
     val rhs = (fmap :@ flatten) :@@ flatten
     //        println(s"check associativity laws for flatten = ${flatten.prettyPrint}:\n\tlhs = ${lhs.simplify.prettyRenamePrint}\n\trhs = ${rhs.simplify.prettyRenamePrint}")
-    lhs equiv rhs
+    TermExpr.extEquals(lhs, rhs)
   }
 
   def checkPureFlattenLaws(fmap: TermExpr, pure: TermExpr, flatten: TermExpr): Boolean = {
@@ -23,7 +23,7 @@ object LawChecking {
     val fpf = (fmap :@ pure) :@@ flatten
 
     //    println(s"check identity laws for pure = ${pure.prettyPrint} and flatten = ${flatten.prettyPrint}:\n\tlhs1 = ${pf.simplify.prettyPrint}\n\trhs1 = ${idFA.simplify.prettyPrint}\n\tlhs2 = ${fpf.simplify.prettyPrint}\n\trhs2 = ${idFA.simplify.prettyPrint}")
-    (pf equiv idFA) && (fpf equiv idFA)
+    TermExpr.extEquals(pf, idFA) && TermExpr.extEquals(fpf, idFA)
   }
 
 }
diff --git a/src/test/scala/io/chymyst/ch/unit/LJTSpec4.scala b/src/test/scala/io/chymyst/ch/unit/LJTSpec4.scala
index 0e2edc5..6fa78c2 100644
--- a/src/test/scala/io/chymyst/ch/unit/LJTSpec4.scala
+++ b/src/test/scala/io/chymyst/ch/unit/LJTSpec4.scala
@@ -1,7 +1,7 @@
 package io.chymyst.ch.unit
 
 import io.chymyst.ch._
-import io.chymyst.ch.data.{LawChecking => LC}
+import io.chymyst.ch.data.{SymbolicLawChecking => LC}
 import org.scalatest.{FlatSpec, Matchers}
 
 class LJTSpec4 extends FlatSpec with Matchers {
diff --git a/src/test/scala/io/chymyst/ch/unit/LambdaTermsSpec.scala b/src/test/scala/io/chymyst/ch/unit/LambdaTermsSpec.scala
index e5aa48e..1e9953d 100644
--- a/src/test/scala/io/chymyst/ch/unit/LambdaTermsSpec.scala
+++ b/src/test/scala/io/chymyst/ch/unit/LambdaTermsSpec.scala
@@ -1,7 +1,7 @@
 package io.chymyst.ch.unit
 
 import io.chymyst.ch._
-import io.chymyst.ch.data.{LawChecking => LC}
+import io.chymyst.ch.data.{SymbolicLawChecking => LC}
 import org.scalatest.{Assertion, FlatSpec, Matchers}
 
 class LambdaTermsSpec extends FlatSpec with Matchers {
@@ -658,7 +658,7 @@ class LambdaTermsSpec extends FlatSpec with Matchers {
     println("Good monads:")
     println(goodMonads.map { case (pure, ftn) ⇒ s"pure = ${pure.prettyPrint}, flatten = ${ftn.prettyPrint}" }.mkString("\n"))
 
-    goodSemimonads.size shouldEqual 2
+    goodSemimonads.size shouldEqual 3
     goodMonads.size shouldEqual 1
   }
 
@@ -718,7 +718,7 @@ class LambdaTermsSpec extends FlatSpec with Matchers {
     println("Good monads:")
     println(goodMonads.map { case (pure, ftn) ⇒ s"pure = ${pure.prettyPrint}, flatten = ${ftn.prettyPrint}" }.mkString("\n"))
 
-    goodSemimonads.size shouldEqual 13
+    goodSemimonads.size shouldEqual 19
     goodMonads.size shouldEqual 6
   }
 
@@ -758,7 +758,7 @@ class LambdaTermsSpec extends FlatSpec with Matchers {
     println("Good monads:")
     println(goodMonads.map { case (pure, ftn) ⇒ s"pure = ${pure.prettyPrint}, flatten = ${ftn.prettyPrint}" }.mkString("\n"))
 
-    goodSemimonads.size shouldEqual 2
+    goodSemimonads.size shouldEqual 3
     goodMonads.size shouldEqual 1
   }
 }
diff --git a/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala b/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala
index 33fe584..5fc9888 100644
--- a/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala
+++ b/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala
@@ -32,7 +32,10 @@ class TermExprSpec extends FlatSpec with Matchers {
       "(0 + Right((0 + Some(Tuple2((0 + Some(z)), (None() + 0))))))",
       "(0 + Right((0 + Some(Tuple2((0 + Some(z)), (0 + Some(z)))))))"
     )
+  }
 
+  it should "compute extensional equality of functions" in {
+    TermExpr.extEquals(TermExpr.id(typeExpr[Int]), TermExpr.id(typeExpr[Int])) shouldEqual true
   }
 
   it should "compute identity function" in {

From b22f039f35c4d467499ea7ae901a0ab87f63a13b Mon Sep 17 00:00:00 2001
From: Sergei Winitzki <winitzki@gmail.com>
Date: Wed, 4 Apr 2018 14:38:32 -0700
Subject: [PATCH 06/11] WIP fixing the bugs with subst and extensional equality

---
 src/main/scala/io/chymyst/ch/TermExpr.scala   | 40 +++++---
 .../chymyst/ch/data/SymbolicLawChecking.scala |  4 +-
 .../io/chymyst/ch/unit/LambdaTermsSpec.scala  | 93 ++++++++++++++++++-
 .../io/chymyst/ch/unit/TermExprSpec.scala     |  4 +-
 4 files changed, 121 insertions(+), 20 deletions(-)

diff --git a/src/main/scala/io/chymyst/ch/TermExpr.scala b/src/main/scala/io/chymyst/ch/TermExpr.scala
index 72adfe2..56f1cea 100644
--- a/src/main/scala/io/chymyst/ch/TermExpr.scala
+++ b/src/main/scala/io/chymyst/ch/TermExpr.scala
@@ -145,7 +145,7 @@ object TermExpr {
     thisVar.name === otherVar.name && (thisVar.t === otherVar.t || TypeExpr.isDisjunctionPart(thisVar.t, otherVar.t))
   }
 
-  /** Replace all non-free occurrences of variable `replaceVar` by expression `byExpr` in `origExpr`.
+  /** Replace all free occurrences of variable `replaceVar` by expression `byExpr` in `origExpr`.
     *
     * @param replaceVar A variable that may occur freely in `origExpr`.
     * @param byExpr     A new expression to replace all free occurrences of that variable.
@@ -156,7 +156,7 @@ object TermExpr {
     // Check that all instances of replaceVar in origExpr have the correct type.
     val badVars = origExpr.freeVars.filter(_.name === replaceVar.name).filterNot(varMatchesType(_, replaceVar))
     if (badVars.nonEmpty) {
-      throw new Exception(s"In subst($replaceVar, $byExpr, $origExpr), found variable(s) ${badVars.map(v ⇒ s"(${v.name}:${v.t.prettyPrint})").mkString(", ")} with incorrect type(s), expected variable type ${replaceVar.t.prettyPrint}")
+      throw new Exception(s"In subst($replaceVar:${replaceVar.t.prettyPrint}, $byExpr, $origExpr), found variable(s) ${badVars.map(v ⇒ s"(${v.name}:${v.t.prettyPrint})").mkString(", ")} with incorrect type(s), expected variable type ${replaceVar.t.prettyPrint}")
     }
     // Do we need an alpha-conversion? Better be safe than sorry.
     val (convertedReplaceVar, convertedOrigExpr) = if (byExpr.usedVarNames contains replaceVar.name) {
@@ -167,7 +167,7 @@ object TermExpr {
 
     substMap(convertedOrigExpr) {
       case c@CurriedE(heads, _) if heads.exists(_.name === convertedReplaceVar.name) ⇒ c
-      // If a variable from `heads` collides with `convertedReplaceVar`, we do not replace anything in the body.
+      // If a variable from `heads` collides with `convertedReplaceVar`, we do not replace anything in the body because the variable occurs as non-free.
 
       case v@VarE(_, _) if varMatchesType(v, convertedReplaceVar) ⇒ byExpr
     }
@@ -361,14 +361,23 @@ object TermExpr {
     * @param termExpr2 The second term.
     * @return `true` if the terms are extensionally equal.
     */
-  def extEquals(termExpr1: TermExpr, termExpr2: TermExpr): Boolean = {
+  def extEqual(termExpr1: TermExpr, termExpr2: TermExpr): Boolean = {
     val t1 = termExpr1.simplify
     val t2 = termExpr2.simplify
-    (t1.t === t2.t) && ((t1 equiv t2) || ((t1, t2) match {
-      case (CurriedE(h1 :: _, _), CurriedE(_ :: _, _)) ⇒
-        subtypeVars(h1.t).forall(term ⇒ extEquals(t1(term), t2(term)))
-      case _ ⇒ false
-    }))
+    (t1.t === t2.t) && (
+      (t1 equiv t2) || {
+        println(s"DEBUG: checking extensional equality of ${t1.prettyPrint} and ${t2.prettyPrint}")
+        (t1, t2) match {
+          case (CurriedE(h1 :: _, _), CurriedE(_ :: _, _)) ⇒
+            subtypeVars(h1.t).forall { term ⇒
+              val result = extEqual(t1(term), t2(term))
+              if (!result) println(s"DEBUG: found inequality after substituting term ${term.prettyPrint}")
+              result
+            }
+          case _ ⇒ false
+        }
+      }
+      )
   }
 }
 
@@ -540,7 +549,7 @@ sealed trait TermExpr {
     letter ← ('a' to 'z').toIterator
   } yield s"$letter$number"
 
-  private lazy val renameBoundVars: TermExpr = TermExpr.substMap(this) {
+  private[ch] lazy val renameBoundVars: TermExpr = TermExpr.substMap(this) {
     case CurriedE(heads, body) ⇒
       val oldAndNewVars = heads.map { v ⇒ (v, VarE(TermExpr.freshIdents(), v.t)) }
       val renamedBody = oldAndNewVars.foldLeft(body.renameBoundVars) { case (prev, (oldVar, newVar)) ⇒
@@ -855,19 +864,22 @@ final case class MatchE(term: TermExpr, cases: List[TermExpr]) extends TermExpr
         // Replace redundant matches on the same term, can be simplified by eliminating one match subexpresssion.
         // Example: q match { case x ⇒ q match { case y ⇒ b; case other ⇒ ... } ... }
         // We already know that q was matched as Left(x). Therefore, we can replace y by x in b and remove the `case other` clause altogether.
-
-        val casesSimplified = cases.zipWithIndex.map { case (c@CurriedE(List(headVar), _), i) ⇒
-          TermExpr.substMap(c.simplifyOnce(withEta)) {
+        // Doing a .renameBoundVars on the cases leads to infinite loops somewhere due to incorrect alpha-conversion.
+        val casesSimplified = cases.map(_.simplifyOnce(withEta))
+          /*
+          .zipWithIndex.map { case (c@CurriedE(List(headVar), _), i) ⇒
+          TermExpr.substMap(c) {
             case MatchE(otherTerm, otherCases) if otherTerm === termSimplified ⇒
               // We already matched `otherTerm`, and we are now in case `c`, which is `case x ⇒ ...`.
               // Therefore we can discard any of the `otherCases` except the one corresponding to `c`.
               // We can replace the `q match { case y ⇒ b; ...}` by `b` after replacing `x` by `y` in `b`.
               val remainingCase = otherCases(i)
               val result = AppE(remainingCase, headVar).simplifyOnce(withEta)
-              println(s"DEBUG: replacing ${MatchE(otherTerm, otherCases)} by $result in ${c.simplifyOnce(withEta)}")
+              //            println(s"DEBUG: replacing ${MatchE(otherTerm, otherCases)} by $result in ${c.simplifyOnce(withEta)}")
               result
           }
         }
+        */
         if (casesSimplified.nonEmpty && {
           casesSimplified.zipWithIndex.forall {
             // Detect a ⇒ a pattern
diff --git a/src/main/scala/io/chymyst/ch/data/SymbolicLawChecking.scala b/src/main/scala/io/chymyst/ch/data/SymbolicLawChecking.scala
index f1a655b..6949d33 100644
--- a/src/main/scala/io/chymyst/ch/data/SymbolicLawChecking.scala
+++ b/src/main/scala/io/chymyst/ch/data/SymbolicLawChecking.scala
@@ -9,7 +9,7 @@ object SymbolicLawChecking {
     val lhs = flatten :@@ flatten
     val rhs = (fmap :@ flatten) :@@ flatten
     //        println(s"check associativity laws for flatten = ${flatten.prettyPrint}:\n\tlhs = ${lhs.simplify.prettyRenamePrint}\n\trhs = ${rhs.simplify.prettyRenamePrint}")
-    TermExpr.extEquals(lhs, rhs)
+    TermExpr.extEqual(lhs, rhs)
   }
 
   def checkPureFlattenLaws(fmap: TermExpr, pure: TermExpr, flatten: TermExpr): Boolean = {
@@ -23,7 +23,7 @@ object SymbolicLawChecking {
     val fpf = (fmap :@ pure) :@@ flatten
 
     //    println(s"check identity laws for pure = ${pure.prettyPrint} and flatten = ${flatten.prettyPrint}:\n\tlhs1 = ${pf.simplify.prettyPrint}\n\trhs1 = ${idFA.simplify.prettyPrint}\n\tlhs2 = ${fpf.simplify.prettyPrint}\n\trhs2 = ${idFA.simplify.prettyPrint}")
-    TermExpr.extEquals(pf, idFA) && TermExpr.extEquals(fpf, idFA)
+    TermExpr.extEqual(pf, idFA) && TermExpr.extEqual(fpf, idFA)
   }
 
 }
diff --git a/src/test/scala/io/chymyst/ch/unit/LambdaTermsSpec.scala b/src/test/scala/io/chymyst/ch/unit/LambdaTermsSpec.scala
index 1e9953d..ebef4ac 100644
--- a/src/test/scala/io/chymyst/ch/unit/LambdaTermsSpec.scala
+++ b/src/test/scala/io/chymyst/ch/unit/LambdaTermsSpec.scala
@@ -602,7 +602,7 @@ class LambdaTermsSpec extends FlatSpec with Matchers {
 
     // Compute flatten terms from flm terms
     val ftnTerms = flmTerms.map(flm ⇒ (flm :@ (px =>: px)).simplify)
-    if (debug) println(s"flatten terms: ${ftnTerms.map(_.prettyPrint)}")
+    if (debug) println(s"flatten terms (type ${ftnTerms.head.t.prettyPrint}):\n\t${ftnTerms.map(_.prettyPrint).mkString("\n\t")}")
 
     val pureTerms = TheoremProver.findProofs(pureVar.t)._2
     if (debug) println(s"pure terms: ${pureTerms.map(_.prettyPrint)}")
@@ -671,7 +671,7 @@ class LambdaTermsSpec extends FlatSpec with Matchers {
 
     def pure[A] = freshVar[A ⇒ P[A]]
 
-    val (goodSemimonads, goodMonads) = semimonadsAndMonads(fmapTerm, pure, flm)
+    val (goodSemimonads, goodMonads) = semimonadsAndMonads(fmapTerm, pure, flm, debug = true)
 
     println(s"Good semimonads (flatten):\n${goodSemimonads.map(_.prettyPrint).mkString("\n")}")
 
@@ -742,6 +742,26 @@ class LambdaTermsSpec extends FlatSpec with Matchers {
     goodMonads.size shouldEqual 2
   }
 
+  it should "check A x A + A x A x A monad" in {
+    type P[A] = Either[(A, A), (A, A, A)]
+
+    def fmapTerm[A, B] = ofType[(A ⇒ B) ⇒ P[A] ⇒ P[B]].lambdaTerm
+
+    def flm[A, B] = freshVar[(A ⇒ P[B]) ⇒ P[A] ⇒ P[B]]
+
+    def pure[A] = freshVar[A ⇒ P[A]]
+
+    val (goodSemimonads, goodMonads) = semimonadsAndMonads(fmapTerm, pure, flm)
+
+    println(s"Good semimonads (flatten):\n${goodSemimonads.map(_.prettyPrint).mkString("\n")}")
+
+    println("Good monads:")
+    println(goodMonads.map { case (pure, ftn) ⇒ s"pure = ${pure.prettyPrint}, flatten = ${ftn.prettyPrint}" }.mkString("\n"))
+
+    goodSemimonads.size shouldEqual 12
+    goodMonads.size shouldEqual 2
+  }
+
   it should "check A + (1 ⇒ A) monad" in {
     type P[A] = Either[A, Unit ⇒ A]
 
@@ -761,4 +781,73 @@ class LambdaTermsSpec extends FlatSpec with Matchers {
     goodSemimonads.size shouldEqual 3
     goodMonads.size shouldEqual 1
   }
+
+  it should "check the 1 + A x A monad by hand" in {
+    type P[A] = Option[(A, A)]
+
+    def fmap[A, B] = ofType[(A ⇒ B) ⇒ P[A] ⇒ P[B]].lambdaTerm
+
+    fmap.prettyPrint shouldEqual "a ⇒ b ⇒ b match { c ⇒ (None() + 0); d ⇒ (0 + Some(Tuple2(a d.value._1, a d.value._2))) }"
+
+    def pure[A] = ofType[A ⇒ P[A]].lambdaTerm
+
+    pure.prettyPrint shouldEqual "a ⇒ (0 + Some(Tuple2(a, a)))"
+
+    // Construct flatten by hand.
+    def ppa[A] = freshVar[P[P[A]]]
+
+    def pa[A] = freshVar[P[A]]
+
+    val none = freshVar[None.type]
+    val ppaNone = ppa(none)
+    val paNone = pa(none)
+    ppaNone.t.prettyPrint shouldEqual "Option[Tuple2[Option[Tuple2[A,A]],Option[Tuple2[A,A]]]]"
+    ppa.t.prettyPrint shouldEqual ppaNone.t.prettyPrint
+
+    def tuplePa[A] = freshVar[Some[(P[A], P[A])]]
+
+    def tupleA0[A] = freshVar[Some[(A, A)]]
+
+    def tupleA1[A] = freshVar[Some[(A, A)]]
+
+    def tupleA2[A] = freshVar[Some[(A, A)]]
+
+    val ftn = (
+      ppa =>: ppa.cases(
+        none =>: paNone,
+        tuplePa =>: tuplePa("value")(0).cases(
+          none =>: paNone,
+          tupleA0 =>: tuplePa("value")(1).cases(
+            none =>: paNone,
+            tupleA1 =>:
+              pa(tupleA0(tupleA0("value").t(tupleA0("value")(0), tupleA1("value")(1))))
+          )
+        )
+      )
+      ).prettyRename
+    ftn.t.prettyPrint shouldEqual "Option[Tuple2[Option[Tuple2[A,A]],Option[Tuple2[A,A]]]] ⇒ Option[Tuple2[A,A]]"
+    ftn.prettyPrint shouldEqual "a ⇒ a match { b ⇒ (b + 0); c ⇒ c.value._1 match { d ⇒ (d + 0); e ⇒ c.value._2 match { f ⇒ (f + 0); g ⇒ (0 + Some(Tuple2(e.value._1, g.value._2))) } } }"
+    // Fails due to violating laws.
+    LC.checkPureFlattenLaws(fmap, pure, ftn) shouldEqual true
+    LC.checkFlattenAssociativity(fmap, ftn) shouldEqual false
+
+    val ftn2 = (
+      ppa =>: ppa.cases(
+        none =>: paNone,
+        tuplePa =>: tuplePa("value")(0).cases(
+          none =>: tuplePa("value")(1),
+          tupleA0 =>: tuplePa("value")(1).cases(
+            none =>: tuplePa("value")(0),
+            tupleA1 =>:
+              pa(tupleA0(tupleA0("value").t(tupleA0("value")(0), tupleA1("value")(1))))
+          )
+        )
+      )
+      ).simplify.prettyRename
+    ftn2.t.prettyPrint shouldEqual "Option[Tuple2[Option[Tuple2[A,A]],Option[Tuple2[A,A]]]] ⇒ Option[Tuple2[A,A]]"
+    ftn2.prettyPrint shouldEqual "a ⇒ a match { b ⇒ (b + 0); c ⇒ c.value._1 match { d ⇒ c.value._2; e ⇒ c.value._2 match { f ⇒ c.value._1; g ⇒ (0 + Some(Tuple2(e.value._1, g.value._2))) } } }"
+    LC.checkPureFlattenLaws(fmap, pure, ftn2) shouldEqual true
+    LC.checkFlattenAssociativity(fmap, ftn2) shouldEqual false
+  }
+
 }
diff --git a/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala b/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala
index 5fc9888..548388a 100644
--- a/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala
+++ b/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala
@@ -35,7 +35,7 @@ class TermExprSpec extends FlatSpec with Matchers {
   }
 
   it should "compute extensional equality of functions" in {
-    TermExpr.extEquals(TermExpr.id(typeExpr[Int]), TermExpr.id(typeExpr[Int])) shouldEqual true
+    TermExpr.extEqual(TermExpr.id(typeExpr[Int]), TermExpr.id(typeExpr[Int])) shouldEqual true
   }
 
   it should "compute identity function" in {
@@ -65,7 +65,7 @@ class TermExprSpec extends FlatSpec with Matchers {
 
     the[Exception] thrownBy {
       TermExpr.subst(var23, termExpr1, var32 =>: var23.copy(t = TP("1"))) shouldEqual (var32 =>: termExpr1)
-    } should have message "In subst(x2, \\((x2:3) ⇒ (x3:2) ⇒ (x4:1) ⇒ x3), \\((x3:2) ⇒ x2)), found variable(s) (x2:1) with incorrect type(s), expected variable type 3"
+    } should have message "In subst(x2:3, \\((x2:3) ⇒ (x3:2) ⇒ (x4:1) ⇒ x3), \\((x3:2) ⇒ x2)), found variable(s) (x2:1) with incorrect type(s), expected variable type 3"
   }
 
   it should "recover from incorrect substitution" in {

From 3764a93fd1b21a28cfc4ad173efe04d4ede02040 Mon Sep 17 00:00:00 2001
From: Sergei Winitzki <winitzki@gmail.com>
Date: Fri, 11 May 2018 21:55:22 -0700
Subject: [PATCH 07/11] start on printScala implementation

---
 README.md                                     |  2 +
 src/main/scala/io/chymyst/ch/TermExpr.scala   | 51 +++++++++----
 .../io/chymyst/ch/data/CategoryTheory.scala   | 75 +++++++++++++++++++
 .../scala/io/chymyst/ch/data/Monoid.scala     | 17 -----
 .../io/chymyst/ch/unit/TermExprSpec.scala     | 20 +++++
 5 files changed, 133 insertions(+), 32 deletions(-)
 create mode 100644 src/main/scala/io/chymyst/ch/data/CategoryTheory.scala
 delete mode 100644 src/main/scala/io/chymyst/ch/data/Monoid.scala

diff --git a/README.md b/README.md
index 862ac6f..67392a1 100644
--- a/README.md
+++ b/README.md
@@ -146,6 +146,8 @@ See the [youtube presentation](https://youtu.be/cESdgot_ZxY) for more details ab
 
 [This lecture](https://youtu.be/v=KcfD3Iv--UM) is a pedagogical explanation of the Curry-Howard correspondence in the context of functional programming.
 
+See also a [recent presentation at the Haskell User's Group meetup](https://youtu.be/OFBwrMo1ESk).
+
 # Unit tests
 
 `sbt test`
diff --git a/src/main/scala/io/chymyst/ch/TermExpr.scala b/src/main/scala/io/chymyst/ch/TermExpr.scala
index 56f1cea..d9aba93 100644
--- a/src/main/scala/io/chymyst/ch/TermExpr.scala
+++ b/src/main/scala/io/chymyst/ch/TermExpr.scala
@@ -349,8 +349,8 @@ object TermExpr {
     * @return A sequence of [[TermExpr]] values containing the necessary fresh variables.
     */
   def subtypeVars(typeExpr: TypeExpr): Seq[TermExpr] = typeExpr match {
-    case dt@DisjunctT(constructor, typeParams, terms) ⇒ terms.zipWithIndex.flatMap { case (t, i) ⇒ subtypeVars(t).map(v ⇒ DisjunctE(i, terms.length, v, dt)) }
-    case nct@NamedConjunctT(constructor, typeParams, accessors, wrapped) ⇒
+    case dt@DisjunctT(_, _, terms) ⇒ terms.zipWithIndex.flatMap { case (t, i) ⇒ subtypeVars(t).map(v ⇒ DisjunctE(i, terms.length, v, dt)) }
+    case nct@NamedConjunctT(_, _, _, wrapped) ⇒
       TheoremProver.explode(wrapped.map(subtypeVars)).map(NamedConjunctE(_, nct))
     case _ ⇒ Seq(VarE(freshIdents(), typeExpr))
   }
@@ -544,6 +544,27 @@ sealed trait TermExpr {
       "(" + leftZeros.mkString(" + ") + leftZerosString + term.prettyPrintWithParentheses(0) + rightZerosString + rightZeros.mkString(" + ") + ")"
   }
 
+  lazy val printScala: String = printScalaWithTypes()
+
+  private[ch] def printScalaWithTypes(withTypes: Boolean = false): String = this match {
+    case VarE(name, _) ⇒ name + (if (withTypes) ": " + t.prettyPrint else "")
+    case AppE(head, arg) ⇒
+      val h = head.printScalaWithTypes(true)
+      val b = arg.printScalaWithTypes()
+      s"$h($b)"
+    case CurriedE(heads, body) ⇒
+      s"${heads.map(_.printScalaWithTypes(true)).mkString(" ⇒ ")} ⇒ ${body.printScalaWithTypes()}"
+    case UnitE(_) ⇒ "()"
+    case ConjunctE(terms) ⇒ "(" + terms.map(_.printScalaWithTypes()).mkString(", ") + ")"
+    case NamedConjunctE(terms, tExpr) ⇒ if (tExpr.wrapped.isEmpty) tExpr.constructor.toString
+    else s"${tExpr.constructor.toString}(${terms.map(_.printScalaWithTypes()).mkString(", ")})"
+    case ProjectE(index, term) ⇒ term.printScalaWithTypes() + "." + term.accessor(index)
+    case MatchE(term, cases) ⇒
+      term.printScalaWithTypes() + " match { case " + cases.map(_.printScalaWithTypes(true)).mkString("; case ") + " }"
+    case DisjunctE(index, total, term, _) ⇒
+      term.printScalaWithTypes()
+  }
+
   private def prettyVars: Iterator[String] = for {
     number ← Iterator.single("") ++ Iterator.from(1).map(_.toString)
     letter ← ('a' to 'z').toIterator
@@ -866,20 +887,20 @@ final case class MatchE(term: TermExpr, cases: List[TermExpr]) extends TermExpr
         // We already know that q was matched as Left(x). Therefore, we can replace y by x in b and remove the `case other` clause altogether.
         // Doing a .renameBoundVars on the cases leads to infinite loops somewhere due to incorrect alpha-conversion.
         val casesSimplified = cases.map(_.simplifyOnce(withEta))
-          /*
-          .zipWithIndex.map { case (c@CurriedE(List(headVar), _), i) ⇒
-          TermExpr.substMap(c) {
-            case MatchE(otherTerm, otherCases) if otherTerm === termSimplified ⇒
-              // We already matched `otherTerm`, and we are now in case `c`, which is `case x ⇒ ...`.
-              // Therefore we can discard any of the `otherCases` except the one corresponding to `c`.
-              // We can replace the `q match { case y ⇒ b; ...}` by `b` after replacing `x` by `y` in `b`.
-              val remainingCase = otherCases(i)
-              val result = AppE(remainingCase, headVar).simplifyOnce(withEta)
-              //            println(s"DEBUG: replacing ${MatchE(otherTerm, otherCases)} by $result in ${c.simplifyOnce(withEta)}")
-              result
-          }
+        /*
+        .zipWithIndex.map { case (c@CurriedE(List(headVar), _), i) ⇒
+        TermExpr.substMap(c) {
+          case MatchE(otherTerm, otherCases) if otherTerm === termSimplified ⇒
+            // We already matched `otherTerm`, and we are now in case `c`, which is `case x ⇒ ...`.
+            // Therefore we can discard any of the `otherCases` except the one corresponding to `c`.
+            // We can replace the `q match { case y ⇒ b; ...}` by `b` after replacing `x` by `y` in `b`.
+            val remainingCase = otherCases(i)
+            val result = AppE(remainingCase, headVar).simplifyOnce(withEta)
+            //            println(s"DEBUG: replacing ${MatchE(otherTerm, otherCases)} by $result in ${c.simplifyOnce(withEta)}")
+            result
         }
-        */
+      }
+      */
         if (casesSimplified.nonEmpty && {
           casesSimplified.zipWithIndex.forall {
             // Detect a ⇒ a pattern
diff --git a/src/main/scala/io/chymyst/ch/data/CategoryTheory.scala b/src/main/scala/io/chymyst/ch/data/CategoryTheory.scala
new file mode 100644
index 0000000..543d660
--- /dev/null
+++ b/src/main/scala/io/chymyst/ch/data/CategoryTheory.scala
@@ -0,0 +1,75 @@
+package io.chymyst.ch.data
+
+// Declarations of standard type classes, to be used in macros.
+
+trait Semigroup[T] {
+  def combine(x: T, y: T): T
+}
+
+trait Monoid[T] extends Semigroup[T] {
+  def empty: T
+}
+
+object Monoid {
+  def empty[T](implicit ev: Monoid[T]): T = ev.empty
+
+  implicit class MonoidSyntax[T](t: T)(implicit ev: Monoid[T]) {
+
+    def combine(y: T): T = ev.combine(t, y)
+  }
+
+}
+
+trait Functor[F[_]] {
+  def map[A, B](fa: F[A])(f: A ⇒ B): F[B]
+}
+
+trait ContraFunctor[F[_]] {
+  def map[A, B](fa: F[A])(f: B ⇒ A): F[B]
+}
+
+trait Filterable[F[_]] extends Functor[F] {
+  def deflate[A](fa: F[Option[A]]): F[A]
+}
+
+trait ContraFilterable[F[_]] extends ContraFunctor[F] {
+  def inflate[A](fa: F[A]): F[Option[A]]
+}
+
+trait Semimonad[F[_]] extends Functor[F] {
+  def join[A](ffa: F[F[A]]): F[A]
+}
+
+trait Pointed[F[_]] extends Functor[F] {
+  def point[A]: F[A]
+}
+
+trait Zippable[F[_]] extends Functor[F] {
+  def zip[A, B](fa: F[A], fb: F[B]): F[(A, B)]
+}
+
+trait Foldable[F[_]] extends Functor[F] {
+  def foldMap[A, B: Monoid](fa: F[A])(f: A ⇒ B)
+}
+
+trait Traversable[F[_]] extends Functor[F] {
+  def sequence[Z[_] : Zippable, A](fga: F[Z[A]]): Z[F[A]]
+}
+
+trait Monad[F[_]] extends Pointed[F] with Semimonad[F]
+
+trait Applicative[F[_]] extends Pointed[F] with Zippable[F]
+
+trait Cosemimonad[F[_]] extends Functor[F] {
+  def cojoin[A](fa: F[A]): F[F[A]]
+}
+
+trait Copointed[F[_]] extends Functor[F] {
+  def extract[A](fa: F[A]): A
+}
+
+trait Comonad[F[_]] extends Copointed[F] with Cosemimonad[F]
+
+trait Cozippable[F[_]] extends Functor[F] {
+  def decide[A, B](fab: F[Either[A, B]]): Either[F[A], F[B]]
+}
diff --git a/src/main/scala/io/chymyst/ch/data/Monoid.scala b/src/main/scala/io/chymyst/ch/data/Monoid.scala
deleted file mode 100644
index dba460d..0000000
--- a/src/main/scala/io/chymyst/ch/data/Monoid.scala
+++ /dev/null
@@ -1,17 +0,0 @@
-package io.chymyst.ch.data
-
-trait Monoid[T] {
-  def empty: T
-
-  def combine(x: T, y: T): T
-}
-
-object Monoid {
-  def empty[T](implicit ev: Monoid[T]): T = ev.empty
-
-  implicit class MonoidSyntax[T](t: T)(implicit ev: Monoid[T]) {
-
-    def combine(y: T): T = ev.combine(t, y)
-  }
-
-}
diff --git a/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala b/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala
index 548388a..7510763 100644
--- a/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala
+++ b/src/test/scala/io/chymyst/ch/unit/TermExprSpec.scala
@@ -32,8 +32,19 @@ class TermExprSpec extends FlatSpec with Matchers {
       "(0 + Right((0 + Some(Tuple2((0 + Some(z)), (None() + 0))))))",
       "(0 + Right((0 + Some(Tuple2((0 + Some(z)), (0 + Some(z)))))))"
     )
+
+    val subtypeVarsScala = TermExpr.subtypeVars(p.t).map(_.printScala).map(_.replaceAll("z[0-9]+", "z")) shouldEqual
+      List("Left(None)", "Left(Some(None))", "Left(Some(Some(Tuple2(z, z))))", "Right(None)", "Right(Some(Tuple2(None, None)))", "Right(Some(Tuple2(None, Some(z))))", "Right(Some(Tuple2(Some(z), None)))", "Right(Some(Tuple2(Some(z), Some(z))))")
   }
 
+  it should "compute Scala code of flatten for Option" in {
+    def flattenOpt[A]: Option[Option[A]] ⇒ Option[A] = implement
+    
+    val flattenScala = flattenOpt.lambdaTerm.printScala
+    
+    flattenScala shouldEqual "a: Option[Option[A]] ⇒ a match { case b: None.type ⇒ None; case c: Some[Option[A]] ⇒ c.value }"
+  }
+  
   it should "compute extensional equality of functions" in {
     TermExpr.extEqual(TermExpr.id(typeExpr[Int]), TermExpr.id(typeExpr[Int])) shouldEqual true
   }
@@ -82,6 +93,15 @@ class TermExprSpec extends FlatSpec with Matchers {
       have message "Incorrect substitution of bound variable x2 by non-variable Tuple2(x, x) in substMap(x2 ⇒ x1){...}"
   }
 
+  behavior of "printScala"
+
+  it should "print functions in Scala syntax" in {
+    termExpr1.printScala shouldEqual "x2 ⇒ x3 ⇒ x4 ⇒ x3"
+    termExpr2.printScala shouldEqual "x2 ⇒ x3 ⇒ x4 ⇒ x1"
+    termExpr3.printScala shouldEqual "x1 ⇒ x2 ⇒ x3 ⇒ x4 ⇒ x1"
+
+  }
+
   behavior of "TermExpr#renameVar"
 
   it should "rename one variable" in {

From d8d666a4f301f47324e2b34baafd6916a7235432 Mon Sep 17 00:00:00 2001
From: "sergei.winitzki" <sergei.winitzki@workday.com>
Date: Mon, 23 Aug 2021 15:38:10 +0200
Subject: [PATCH 08/11] add some build.sbt settings so they don't get lost

---
 build.sbt | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/build.sbt b/build.sbt
index ecd1f19..2376f53 100644
--- a/build.sbt
+++ b/build.sbt
@@ -157,14 +157,14 @@ lazy val curryhoward: Project = (project in file("."))
 /////////////////////////////////////////////////////////////////////////////////////////////////////
 // Publishing to Sonatype Maven repository
 publishMavenStyle := true
-publishTo := sonatypePublishToBundle.value
-/*{
+publishTo := //sonatypePublishToBundle.value
+{
   val nexus = "https://oss.sonatype.org/"
   if (isSnapshot.value)
     Some("snapshots" at nexus + "content/repositories/snapshots")
   else
     Some("releases" at nexus + "service/local/staging/deploy/maven2")
-}*/
+}
 //
 publishArtifact in Test := false
 //

From 7d7c8a39ad1fe9728ec154a4c1b2baba0ae963f0 Mon Sep 17 00:00:00 2001
From: "sergei.winitzki" <sergei.winitzki@workday.com>
Date: Sun, 29 Aug 2021 21:05:30 +0200
Subject: [PATCH 09/11] update READMe and bump version

---
 README.md | 1 +
 build.sbt | 2 +-
 2 files changed, 2 insertions(+), 1 deletion(-)

diff --git a/README.md b/README.md
index ac7ca6e..c8a87a0 100644
--- a/README.md
+++ b/README.md
@@ -158,6 +158,7 @@ Build the tutorial (thanks to the [tut plugin](https://github.com/tpolecat/tut))
 
 # Revision history
 
+- 0.3.8 Support Scala 2.13 (keep supporting Scala 2.11 and 2.12)
 - 0.3.7 Implement the `typeExpr` macro instead of the old test-only API. Detect and use `val`s from the immediately enclosing class. Minor performance improvements and bug fixes (alpha-conversion for STLC terms). Tests for automatic discovery of some monads.
 - 0.3.6 STLC terms are now emitted for `implement` as well; the JVM bytecode limit is obviated; fixed bug with recognizing `Function10`.
 - 0.3.5 Added `:@@` and `@@:` operations to the STLC interpreter. Fixed a bug whereby `Tuple2(x._1, x._2)` was not simplified to `x`. Fixed other bugs in alpha-conversion of type parameters.
diff --git a/build.sbt b/build.sbt
index 2376f53..5dd9f8a 100644
--- a/build.sbt
+++ b/build.sbt
@@ -124,7 +124,7 @@ lazy val curryhoward: Project = (project in file("."))
   .settings(common)
   .settings(
     organization := "io.chymyst",
-    version := "0.3.8",
+    version := "0.3.9",
 
     licenses := Seq("Apache License, Version 2.0" -> url("https://www.apache.org/licenses/LICENSE-2.0.txt")),
     homepage := Some(url("https://github.com/Chymyst/curryhoward")),

From 4f036347aa256104550a85b22331400e0e4a816c Mon Sep 17 00:00:00 2001
From: "sergei.winitzki" <sergei.winitzki@workday.com>
Date: Sat, 6 Nov 2021 12:51:59 +0100
Subject: [PATCH 10/11] one more test that fails

---
 src/test/scala/io/chymyst/ch/unit/LJTSpec3.scala | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/src/test/scala/io/chymyst/ch/unit/LJTSpec3.scala b/src/test/scala/io/chymyst/ch/unit/LJTSpec3.scala
index 157f402..48e0bf4 100644
--- a/src/test/scala/io/chymyst/ch/unit/LJTSpec3.scala
+++ b/src/test/scala/io/chymyst/ch/unit/LJTSpec3.scala
@@ -123,6 +123,12 @@ class LJTSpec3 extends FlatSpec with Matchers with BeforeAndAfterEach {
     f3.size shouldEqual 1
   }
 
+  it should "generate methods for State monad with Int state" in {
+    final case class S_int[A](run: Int => (A, Int)) // State monad with internal state of type Int.
+    val wu1: S_int[Unit] = S_int { i => ((), i) }
+    val wu2: S_int[Unit] = implement // Expect the same
+  }
+
   it should "generate methods for Continuation monad with no ambiguity" in {
     case class Cont[X, R](c: (X ⇒ R) ⇒ R)
 

From 4769b4e1b4f6b5913b098b6005ffbe77c8d35c55 Mon Sep 17 00:00:00 2001
From: "sergei.winitzki" <sergei.winitzki@workday.com>
Date: Sat, 6 Nov 2021 12:53:28 +0100
Subject: [PATCH 11/11] add more tests

---
 src/test/scala/io/chymyst/ch/unit/LJTSpec3.scala | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/src/test/scala/io/chymyst/ch/unit/LJTSpec3.scala b/src/test/scala/io/chymyst/ch/unit/LJTSpec3.scala
index 48e0bf4..d426ca2 100644
--- a/src/test/scala/io/chymyst/ch/unit/LJTSpec3.scala
+++ b/src/test/scala/io/chymyst/ch/unit/LJTSpec3.scala
@@ -126,7 +126,9 @@ class LJTSpec3 extends FlatSpec with Matchers with BeforeAndAfterEach {
   it should "generate methods for State monad with Int state" in {
     final case class S_int[A](run: Int => (A, Int)) // State monad with internal state of type Int.
     val wu1: S_int[Unit] = S_int { i => ((), i) }
+    wu1.run(123) shouldEqual (((), 123))
     val wu2: S_int[Unit] = implement // Expect the same
+    wu2.run(123) shouldEqual (((), 123))
   }
 
   it should "generate methods for Continuation monad with no ambiguity" in {