From cbec1558fa1fcbcaa1a5a5f091f4e933021ebf45 Mon Sep 17 00:00:00 2001
From: SimonGuilloud <simon.guilloud@epfl.ch>
Date: Fri, 19 Apr 2024 14:33:58 +0200
Subject: [PATCH] Egraph (#220)

Add a tactic called "Congruence".
This tactic tries to prove a sequent by congruence.
Consider the congruence closure of all terms and formulas in the sequent, with respect to all === and <=> left of the sequent.
The sequent is provable by congruence if one of the following conditions is met:
- The right side contains an equality s === t or equivalence a <=> b provable in the congruence closure.
- The left side contains an negated equality !(s === t) or equivalence !(a <=> b) provable in the congruence closure.
- There is a formula a on the left and b on the right such that a and b are congruent.
- There are two formulas a and !b on the left such that a and b are congruent.
- There are two formulas a and !b on the right such that a and b are congruent.
- The sequent is Ol-valid without equality reasoning
Note that complete congruence closure modulo OL is an open problem.

The tactic uses an egraph datastructure to compute the congruence closure.
The egraph itselfs relies on two underlying union-find datastructure, one for terms and one for formulas.
The union-finds are equiped with an explain method that produces a path between any two elements in the same equivalence class.
Each edge of the path can come from an external equality, or be the consequence of congruence.
The tactic uses uses this path to produce needed proofs.
---
 CHANGES.md                                    |   3 +
 lisa-examples/src/main/scala/Example.scala    |  87 +-
 .../scala/lisa/automation/Congruence.scala    | 624 ++++++++++++
 .../lisa/maths/settheory/SetTheory2.scala     |   4 +-
 .../lisa/automation/CongruenceTest.scala      | 914 ++++++++++++++++++
 .../test/scala/lisa/utilities/TestMain.scala  |  16 +
 .../src/main/scala/lisa/fol/FOLHelpers.scala  |   3 +
 .../main/scala/lisa/prooflib/BasicMain.scala  |   4 +-
 .../scala/lisa/prooflib/ProofsHelpers.scala   |   4 +-
 refman/lisa.pdf                               | Bin 251598 -> 256758 bytes
 refman/tactics.tex                            |  53 +
 11 files changed, 1678 insertions(+), 34 deletions(-)
 create mode 100644 lisa-sets/src/main/scala/lisa/automation/Congruence.scala
 create mode 100644 lisa-sets/src/test/scala/lisa/automation/CongruenceTest.scala
 create mode 100644 lisa-sets/src/test/scala/lisa/utilities/TestMain.scala

diff --git a/CHANGES.md b/CHANGES.md
index 9e00084a..b71a20ae 100644
--- a/CHANGES.md
+++ b/CHANGES.md
@@ -1,5 +1,8 @@
 # Change List
 
+## 2024-04-12
+Addition of the Congruence tactic, solving sequents by congruence closure using egraphs.
+
 ## 2024-04-12
 Addition of simply typed lambda calculus with top level polymorphism and inductive poylmorphic algebraic data types.
 Addition of tactics for typechecking and structural induction over ADTs.
diff --git a/lisa-examples/src/main/scala/Example.scala b/lisa-examples/src/main/scala/Example.scala
index c942aff6..87ec3821 100644
--- a/lisa-examples/src/main/scala/Example.scala
+++ b/lisa-examples/src/main/scala/Example.scala
@@ -1,8 +1,10 @@
 import lisa.automation.Substitution.{ApplyRules as Substitute}
 import lisa.automation.Tableau
 import lisa.automation.atp.Goeland
+import lisa.automation.Congruence
 
 object Example extends lisa.Main {
+  draft()
 
   val x = variable
   val y = variable
@@ -61,7 +63,31 @@ object Example extends lisa.Main {
   }
 
   val buveurs2 = Theorem(exists(x, P(x) ==> forall(y, P(y)))) {
-    have(thesis) by Goeland("goeland/Example.buveurs2_sol")
+    have(thesis) by Goeland//("goeland/Example.buveurs2_sol")
+  }
+
+
+  val a = variable
+  val one = variable
+  val two = variable
+  val * = SchematicFunctionLabel("*", 2)
+  val << = SchematicFunctionLabel("<<", 2)
+  val / = SchematicFunctionLabel("/", 2)
+  private val star: SchematicFunctionLabel[2] = *
+  private val shift: SchematicFunctionLabel[2] = <<
+  private val divide: SchematicFunctionLabel[2] = /
+  
+  while (true) {
+    ()
+  }
+
+  extension (t:Term) {
+    def *(u:Term) = star(t, u)
+    def <<(u:Term) = shift(t, u)
+    def /(u:Term) = divide(t, u)
+  }
+  val congruence = Theorem(((a*two) === (a<<one), a*(two/two) === (a*two)/two, (two/two) === one, (a*one) === a) |- ((a<<one)/two) === a) {
+    have(thesis) by Congruence
   }
 
   /*
@@ -78,40 +104,43 @@ object Example extends lisa.Main {
 
    */
 
-  /*
+  
 
 
 
 // Simple tactic definition for LISA DSL
+/*
 import lisa.automation.kernel.OLPropositionalSolver.*
 
-// object SimpleTautology extends ProofTactic {
-//   def solveFormula(using proof: library.Proof)(f: Formula, decisionsPos: List[Formula], decisionsNeg: List[Formula]): proof.ProofTacticJudgement = {
-//     val redF = reducedForm(f)
-//     if (redF == ⊤) {
-//       Restate(decisionsPos |- f :: decisionsNeg)
-//     } else if (redF == ⊥) {
-//       proof.InvalidProofTactic("Sequent is not a propositional tautology")
-//     } else {
-//       val atom = findBestAtom(redF).get
-//       def substInRedF(f: Formula) = redF.substituted(atom -> f)
-//       TacticSubproof {
-//         have(solveFormula(substInRedF(⊤), atom :: decisionsPos, decisionsNeg))
-//         val step2 = thenHave(atom :: decisionsPos |- redF :: decisionsNeg) by Substitution2(⊤ <=> atom)
-//         have(solveFormula(substInRedF(⊥), decisionsPos, atom :: decisionsNeg))
-//         val step4 = thenHave(decisionsPos |- redF :: atom :: decisionsNeg) by Substitution2(⊥ <=> atom)
-//         have(decisionsPos |- redF :: decisionsNeg) by Cut(step4, step2)
-//         thenHave(decisionsPos |- f :: decisionsNeg) by Restate
-//       }
-//     }
-//   }
-//   def solveSequent(using proof: library.Proof)(bot: Sequent) =
-//     TacticSubproof { // Since the tactic above works on formulas, we need an extra step to convert an arbitrary sequent to an equivalent formula
-//       have(solveFormula(sequentToFormula(bot), Nil, Nil))
-//       thenHave(bot) by Restate.from
-//     }
-// }
-   */
+object SimpleTautology extends ProofTactic {
+  def solveFormula(using proof: library.Proof)(f: Formula, decisionsPos: List[Formula], decisionsNeg: List[Formula]): proof.ProofTacticJudgement = {
+    val redF = reducedForm(f)
+    if (redF == ⊤) {
+      Restate(decisionsPos |- f :: decisionsNeg)
+    } else if (redF == ⊥) {
+      proof.InvalidProofTactic("Sequent is not a propositional tautology")
+    } else {
+      val atom = findBestAtom(redF).get
+      def substInRedF(f: Formula) = redF.substituted(atom -> f)
+      TacticSubproof {
+        have(solveFormula(substInRedF(⊤), atom :: decisionsPos, decisionsNeg))
+        val step2 = thenHave(atom :: decisionsPos |- redF :: decisionsNeg) by Substitution2(⊤ <=> atom)
+        have(solveFormula(substInRedF(⊥), decisionsPos, atom :: decisionsNeg))
+        val step4 = thenHave(decisionsPos |- redF :: atom :: decisionsNeg) by Substitution2(⊥ <=> atom)
+        have(decisionsPos |- redF :: decisionsNeg) by Cut(step4, step2)
+        thenHave(decisionsPos |- f :: decisionsNeg) by Restate
+      }
+    }
+  }
+
+  def solveSequent(using proof: library.Proof)(bot: Sequent) =
+    TacticSubproof { // Since the tactic above works on formulas, we need an extra step to convert an arbitrary sequent to an equivalent formula
+      have(solveFormula(sequentToFormula(bot), Nil, Nil))
+      thenHave(bot) by Restate.from
+    }
+}
+*/
+  
   // val a = formulaVariable()
   // val b = formulaVariable()
   // val c = formulaVariable()
diff --git a/lisa-sets/src/main/scala/lisa/automation/Congruence.scala b/lisa-sets/src/main/scala/lisa/automation/Congruence.scala
new file mode 100644
index 00000000..ccf25295
--- /dev/null
+++ b/lisa-sets/src/main/scala/lisa/automation/Congruence.scala
@@ -0,0 +1,624 @@
+package lisa.automation
+import lisa.fol.FOL.{*, given}
+import lisa.prooflib.BasicStepTactic.*
+import lisa.prooflib.ProofTacticLib.*
+import lisa.prooflib.SimpleDeducedSteps.*
+import lisa.prooflib.*
+import lisa.utils.parsing.UnreachableException
+import leo.datastructures.TPTP.CNF.AtomicFormula
+
+/**
+  * This tactic tries to prove a sequent by congruence.
+  * Consider the congruence closure of all terms and formulas in the sequent, with respect to all === and <=> left of the sequent.
+  * The sequent is provable by congruence if one of the following conditions is met:
+  *    - The right side contains an equality s === t or equivalence a <=> b provable in the congruence closure.
+  *    - The left side contains an negated equality !(s === t) or equivalence !(a <=> b) provable in the congruence closure.
+  *    - There is a formula a on the left and b on the right such that a and b are congruent.
+  *    - There are two formulas a and !b on the left such that a and b are congruent.
+  *    - There are two formulas a and !b on the right such that a and b are congruent.
+  *    - The sequent is Ol-valid without equality reasoning
+  * Note that complete congruence closure modulo OL is an open problem.
+  * 
+  * The tactic uses an egraph datastructure to compute the congruence closure.
+  * The egraph itselfs relies on two underlying union-find datastructure, one for terms and one for formulas.
+  * The union-finds are equiped with an `explain` method that produces a path between any two elements in the same equivalence class.
+  * Each edge of the path can come from an external equality, or be the consequence of congruence.
+  * The tactic uses uses this path to produce needed proofs.
+  * 
+  */
+object Congruence  extends ProofTactic with ProofSequentTactic {
+    def apply(using lib: Library, proof: lib.Proof)(bot: Sequent): proof.ProofTacticJudgement = TacticSubproof {
+      import lib.*
+
+      val egraph = new EGraphTerms()
+      egraph.addAll(bot.left)
+      egraph.addAll(bot.right)
+
+      bot.left.foreach{
+        case (left === right) => egraph.merge(left, right)
+        case (left <=> right) => egraph.merge(left, right)
+        case _ => ()
+      }
+
+      if isSameSequent(bot, ⊤) then
+        have(bot) by Restate
+      else if bot.left.exists { lf =>
+        bot.right.exists { rf =>
+          if egraph.idEq(lf, rf) then
+            val base = have(bot.left |- (bot.right + lf) ) by Restate
+            val eq = have(egraph.proveFormula(lf, rf, bot))
+            val a = formulaVariable
+            have((bot.left + (lf <=> rf)) |- (bot.right) ) by RightSubstIff.withParametersSimple(List((lf, rf)), lambda(a, a))(base)
+            have(bot) by Cut(eq, lastStep)
+            true
+          else false
+        } ||
+        bot.left.exists{ 
+          case rf2 @ Neg(rf) if egraph.idEq(lf, rf)=>
+            val base = have((bot.left + !lf) |- bot.right ) by Restate
+            val eq = have(egraph.proveFormula(lf, rf, bot))
+            val a = formulaVariable
+            have((bot.left + (lf <=> rf)) |- (bot.right) ) by LeftSubstIff.withParametersSimple(List((lf, rf)), lambda(a, !a))(base)
+            have(bot) by Cut(eq, lastStep)
+            true
+          case _  => false
+        } || {
+        lf match
+          case !(a === b) if egraph.idEq(a, b) => 
+            have(egraph.proveTerm(a, b, bot))
+            true
+          case !(a <=> b) if egraph.idEq(a, b) => 
+            have(egraph.proveFormula(a, b, bot))
+            true
+          case _ => false
+        }
+
+      } then ()
+      else if bot.right.exists { rf =>
+        bot.right.exists{ 
+          case lf2 @ Neg(lf) if egraph.idEq(lf, rf)=>
+            val base = have((bot.left) |- (bot.right + !rf) ) by Restate
+            val eq = have(egraph.proveFormula(lf, rf, bot))
+            val a = formulaVariable
+            have((bot.left + (lf <=> rf)) |- (bot.right) ) by RightSubstIff.withParametersSimple(List((lf, rf)), lambda(a, !a))(base)
+            have(bot) by Cut(eq, lastStep)
+            true
+          case _  => false
+        } || {
+        rf match
+          case (a === b) if egraph.idEq(a, b) => 
+            have(egraph.proveTerm(a, b, bot))
+            true
+          case (a <=> b) if egraph.idEq(a, b) =>
+            have(egraph.proveFormula(a, b, bot))
+            true
+          case _ => false
+        }
+      } then ()
+      else
+        return proof.InvalidProofTactic(s"No congruence found to show sequent\n $bot")
+    }
+
+  
+}
+
+
+class UnionFind[T]  {
+  // parent of each element, leading to its root. Uses path compression
+  val parent = scala.collection.mutable.Map[T, T]()
+  // original parent of each element, leading to its root. Does not use path compression. Used for explain.
+  val realParent = scala.collection.mutable.Map[T, (T, ((T, T), Boolean, Int))]()
+  //keep track of the rank (i.e. number of elements bellow it) of each element. Necessary to optimize union.
+  val rank = scala.collection.mutable.Map[T, Int]()
+  //tracks order of ancientness of unions.
+  var unionCounter = 0
+
+  /**
+    * add a new element to the union-find.
+    */
+  def add(x: T): Unit = {
+    parent(x) = x
+    realParent(x) = (x, ((x, x), true, 0))
+    rank(x) = 0
+  }
+
+  /**
+    *
+    * @param x the element whose parent we want to find
+    * @return the root of x
+    */
+  def find(x: T): T = {
+    if parent(x) == x then
+      x
+    else
+      var root = x
+      while parent(root) != root do
+        root = parent(root)
+      var y = x
+      while parent(y) != root do
+        parent(y) = root
+        y = parent(y)
+      root
+  }
+
+  /**
+    * Merges the classes of x and y
+    */
+  def union(x: T, y: T): Unit = {
+    unionCounter += 1
+    val xRoot = find(x)
+    val yRoot = find(y)
+    if (xRoot == yRoot) return
+    if (rank(xRoot) < rank(yRoot)) {
+      parent(xRoot) = yRoot
+      realParent(xRoot) = (yRoot, ((x, y), true, unionCounter))
+    } else if (rank(xRoot) > rank(yRoot)) {
+      parent(yRoot) = xRoot
+      realParent(yRoot) = (xRoot, ((x, y), false, unionCounter))
+    } else {
+      parent(yRoot) = xRoot
+      realParent(yRoot) = (xRoot, ((x, y), false, unionCounter))
+      rank(xRoot) = rank(xRoot) + 1
+    }
+  }
+
+  private def getPathToRoot(x: T): List[T] = {
+    if x == find(x) then
+      List(x)
+    else
+      val next = realParent(x)
+      x :: getPathToRoot(next._1)
+
+  }
+
+  private def getExplanationFromTo(x:T, c: T): List[(T, ((T, T), Boolean, Int))] = {
+    if x == c then
+      List()
+    else
+      val next = realParent(x)
+      next :: getExplanationFromTo(next._1, c)}
+
+  private def lowestCommonAncestor(x: T, y: T): Option[T] = {
+    val pathX = getPathToRoot(x)
+    val pathY = getPathToRoot(y)
+    pathX.find(pathY.contains)
+  }
+
+  /**
+    * Returns a path from x to y made of pairs of elements (u, v)
+    * such that union(u, v) was called.
+    */
+  def explain(x:T, y:T): Option[List[(T, T)]]= {
+
+    if (x == y) then return Some(List())
+    val lca = lowestCommonAncestor(x, y)
+    lca match
+      case None => None
+      case Some(lca) =>
+        var max :((T, T), Boolean, Int) = ((x, x), true, 0)
+        var itX = x
+        while itX != lca do
+          val (next, ((u1, u2), b, c)) = realParent(itX)
+          if c > max._3 then
+            max = ((u1, u2), b, c)
+          itX = next
+
+        var itY = y
+        while itY != lca do
+          val (next, ((u1, u2), b, c)) = realParent(itY)
+          if c > max._3 then
+            max = ((u1, u2), !b, c)
+          itY = next
+
+        val u1 = max._1._1
+        val u2 = max._1._2
+        if max._2 then
+          Some(explain(x, u1).get ++ List((u1, u2)) ++ explain(u2, y).get)
+        else
+          Some(explain(x, u2).get ++ List((u1, u2)) ++ explain(u1, y).get)
+  }
+
+
+  /**
+    * Returns the set of all roots of all classes
+    */
+  def getClasses: Set[T] = parent.keys.map(find).toSet
+
+  /**
+    * Add all elements in the collection to the union-find
+    */
+  def addAll(xs: Iterable[T]): Unit = xs.foreach(add)
+
+}
+
+
+///////////////////////////////
+///////// E-graph /////////////
+///////////////////////////////
+
+import scala.collection.mutable
+
+class EGraphTerms() {
+
+  type ENode = Term | Formula
+
+
+
+  val termMap = mutable.Map[Term, Set[Term]]()
+  val termParents = mutable.Map[Term, mutable.Set[AppliedFunctional | AppliedPredicate]]()
+  var termWorklist = List[Term]()
+  val termUF = new UnionFind[Term]()
+
+
+
+
+  val formulaMap = mutable.Map[Formula, Set[Formula]]()
+  val formulaParents = mutable.Map[Formula, mutable.Set[AppliedConnector]]()
+  var formulaWorklist = List[Formula]()
+  val formulaUF = new UnionFind[Formula]()
+
+
+
+
+  trait TermStep
+  case class TermExternal(between: (Term, Term)) extends TermStep
+  case class TermCongruence(between: (Term, Term)) extends TermStep
+
+  trait FormulaStep
+  case class FormulaExternal(between: (Formula, Formula)) extends FormulaStep
+  case class FormulaCongruence(between: (Formula, Formula)) extends FormulaStep
+
+  val termProofMap = mutable.Map[(Term, Term), TermStep]()
+  val formulaProofMap = mutable.Map[(Formula, Formula), FormulaStep]()
+
+  def explain(id1: Term, id2: Term): Option[List[TermStep]] = {
+    val steps = termUF.explain(id1, id2)
+    steps.map(_.foldLeft((id1, List[TermStep]())) {
+      case ((prev, acc), step) =>
+      termProofMap(step) match
+        case s @ TermExternal((l, r)) => 
+          if l == prev then
+            (r, s :: acc)
+          else if r == prev then
+            (l, TermExternal(r, l) :: acc)
+          else throw new Exception("Invalid proof recovered: It is not a chain")
+        case s @ TermCongruence((l, r)) => 
+          if l == prev then
+            (r, s :: acc)
+          else if r == prev then
+            (l, TermCongruence(r, l) :: acc)
+          else throw new Exception("Invalid proof recovered: It is not a chain")
+
+    }._2.reverse)
+  }
+
+  def explain(id1: Formula, id2: Formula): Option[List[FormulaStep]] = {
+    val steps = formulaUF.explain(id1, id2)
+    steps.map(_.foldLeft((id1, List[FormulaStep]())) {
+      case ((prev, acc), step) =>
+      formulaProofMap(step) match
+        case s @ FormulaExternal((l, r)) => 
+          if l == prev then
+            (r, s :: acc)
+          else if r == prev then
+            (l, FormulaExternal(r, l) :: acc)
+          else throw new Exception("Invalid proof recovered: It is not a chain")
+        case s @ FormulaCongruence((l, r)) => 
+          if l == prev then
+            (r, s :: acc)
+          else if r == prev then
+            (l, FormulaCongruence(r, l) :: acc)
+          else throw new Exception("Invalid proof recovered: It is not a chain")
+
+    }._2.reverse)
+  }
+
+
+  def makeSingletonEClass(node:Term): Term = {
+    termUF.add(node)
+    termMap(node) = Set(node)
+    termParents(node) = mutable.Set()
+    node
+  }
+  def makeSingletonEClass(node:Formula): Formula = {
+    formulaUF.add(node)
+    formulaMap(node) = Set(node)
+    formulaParents(node) = mutable.Set()
+    node
+  }
+
+  def classOf(id: Term): Set[Term] = termMap(id)
+  def classOf(id: Formula): Set[Formula] = formulaMap(id)
+
+  def idEq(id1: Term, id2: Term): Boolean = termUF.find(id1) == termUF.find(id2)
+  def idEq(id1: Formula, id2: Formula): Boolean = formulaUF.find(id1) == formulaUF.find(id2)
+
+  def canonicalize(node: Term): Term = node match
+    case AppliedFunctional(label, args) => 
+      AppliedFunctional(label, args.map(termUF.find.asInstanceOf))
+    case _ => node
+  
+    
+  def canonicalize(node: Formula): Formula = {
+    node match
+      case AppliedPredicate(label, args) => AppliedPredicate(label, args.map(termUF.find))
+      case AppliedConnector(label, args) => AppliedConnector(label, args.map(formulaUF.find))
+      case node => node
+  }
+
+  def add(node: Term): Term =
+    if termMap.contains(node) then return node
+    makeSingletonEClass(node)
+    node match
+      case node @ AppliedFunctional(_, args) => 
+        args.foreach(child => 
+          add(child)
+          termParents(child).add(node)
+        )
+        node
+      case _ => node
+  
+  def add(node: Formula): Formula =
+    if formulaMap.contains(node) then return node
+    makeSingletonEClass(node)
+    node match
+      case node @ AppliedPredicate(_, args) => 
+        args.foreach(child => 
+          add(child)
+          termParents(child).add(node)
+        )
+        node
+      case node @ AppliedConnector(_, args) =>
+        args.foreach(child => 
+          add(child)
+          formulaParents(child).add(node)
+        )
+        node
+      case _ => node
+
+  def addAll(nodes: Iterable[Term|Formula]): Unit = 
+    nodes.foreach{
+      case node: Term => add(node)
+      case node: Formula => add(node)
+    }
+    
+    
+
+
+  def merge(id1: Term, id2: Term): Unit = {
+    mergeWithStep(id1, id2, TermExternal((id1, id2)))
+  }
+  def merge(id1: Formula, id2: Formula): Unit = {
+    mergeWithStep(id1, id2, FormulaExternal((id1, id2)))
+  }
+
+  protected def mergeWithStep(id1: Term, id2: Term, step: TermStep): Unit = {
+    if termUF.find(id1) == termUF.find(id2) then ()
+    else
+      termProofMap((id1, id2)) = step
+      val newSet = termMap(termUF.find(id1)) ++ termMap(termUF.find(id2))
+      val newparents = termParents(termUF.find(id1)) ++ termParents(termUF.find(id2))
+      termUF.union(id1, id2)
+      val newId1 = termUF.find(id1)
+      val newId2 = termUF.find(id2)
+      termMap(newId1) = newSet
+      termMap(newId2) = newSet
+      termParents(newId1) = newparents
+      termParents(newId2) = newparents
+    
+      val id = termUF.find(id2)
+      termWorklist = id :: termWorklist
+      val cause = (id1, id2)
+      val termSeen = mutable.Map[Term, AppliedFunctional]()
+      val formulaSeen = mutable.Map[Formula, AppliedPredicate]()
+      newparents.foreach { 
+        case pTerm: AppliedFunctional =>
+          val canonicalPTerm = canonicalize(pTerm) 
+          if termSeen.contains(canonicalPTerm) then 
+            val qTerm = termSeen(canonicalPTerm)
+            Some((pTerm, qTerm, cause))
+            mergeWithStep(pTerm, qTerm, TermCongruence((pTerm, qTerm)))
+          else
+            termSeen(canonicalPTerm) = pTerm
+        case pFormula: AppliedPredicate =>
+          val canonicalPFormula = canonicalize(pFormula) 
+          if formulaSeen.contains(canonicalPFormula) then 
+            val qFormula = formulaSeen(canonicalPFormula)
+
+            Some((pFormula, qFormula, cause))
+            mergeWithStep(pFormula, qFormula, FormulaCongruence((pFormula, qFormula)))
+          else
+            formulaSeen(canonicalPFormula) = pFormula
+      }
+      termParents(id) = (termSeen.values.to(mutable.Set): mutable.Set[AppliedFunctional | AppliedPredicate])  ++ formulaSeen.values.to(mutable.Set)
+  }
+
+  protected def mergeWithStep(id1: Formula, id2: Formula, step: FormulaStep): Unit = 
+    if formulaUF.find(id1) == formulaUF.find(id2) then ()
+    else
+      formulaProofMap((id1, id2)) = step
+      val newSet = formulaMap(formulaUF.find(id1)) ++ formulaMap(formulaUF.find(id2))
+      val newparents = formulaParents(formulaUF.find(id1)) ++ formulaParents(formulaUF.find(id2))
+      formulaUF.union(id1, id2)
+      val newId1 = formulaUF.find(id1)
+      val newId2 = formulaUF.find(id2)
+      formulaMap(newId1) = newSet
+      formulaMap(newId2) = newSet
+      formulaParents(newId1) = newparents
+      formulaParents(newId2) = newparents
+      val id = formulaUF.find(id2)
+      formulaWorklist = id :: formulaWorklist
+      val cause = (id1, id2)
+      val formulaSeen = mutable.Map[Formula, AppliedConnector]()
+      newparents.foreach { 
+        case pFormula: AppliedConnector =>
+          val canonicalPFormula = canonicalize(pFormula) 
+          if formulaSeen.contains(canonicalPFormula) then 
+            val qFormula = formulaSeen(canonicalPFormula)
+            Some((pFormula, qFormula, cause))
+            mergeWithStep(pFormula, qFormula, FormulaCongruence((pFormula, qFormula)))
+          else
+            formulaSeen(canonicalPFormula) = pFormula
+      }
+      formulaParents(id) = formulaSeen.values.to(mutable.Set)
+
+
+  def proveTerm(using lib: Library, proof: lib.Proof)(id1: Term, id2:Term, base: Sequent): proof.ProofTacticJudgement = 
+    TacticSubproof { proveInnerTerm(id1, id2, base) }
+
+  def proveInnerTerm(using lib: Library, proof: lib.Proof)(id1: Term, id2:Term, base: Sequent): Unit = {
+    import lib.*
+    val steps = explain(id1, id2)
+    steps match {
+      case None => throw new Exception("No proof found in the egraph")
+      case Some(steps) => 
+        if steps.isEmpty then have(base.left |- (base.right + (id1 === id2))) by Restate
+        steps.foreach {
+          case TermExternal((l, r)) => 
+            val goalSequent = base.left |- (base.right + (id1 === r))
+            if l == id1 then 
+              have(goalSequent) by Restate
+            else
+              val x = freshVariable(id1)
+              have(goalSequent) by RightSubstEq.withParametersSimple(List((l, r)), lambda(x, id1 === x))(lastStep)
+          case TermCongruence((l, r)) => 
+            val prev = if id1 != l then lastStep else null
+            val leqr = have(base.left |- (base.right + (l === r))) subproof { sp ?=>
+              (l, r) match
+                case (AppliedFunctional(labell, argsl), AppliedFunctional(labelr, argsr)) if labell == labelr && argsl.size == argsr.size => 
+                  var freshn = freshId((l.freeVariables ++ r.freeVariables).map(_.id), "n").no
+                  val ziped = (argsl zip argsr)
+                  var zip = List[(Term, Term)]()
+                  var children = List[Term]()
+                  var vars = List[Variable]()
+                  var steps = List[(Formula, sp.ProofStep)]()
+                  ziped.reverse.foreach { (al, ar) =>
+                    if al == ar then children = al :: children
+                    else {
+                      val x = Variable(Identifier("n", freshn))
+                      freshn = freshn + 1
+                      children = x :: children
+                      vars = x :: vars
+                      steps = (al === ar, have(proveTerm(al, ar, base))) :: steps
+                      zip = (al, ar) :: zip
+                    }
+                  }
+                  have(base.left |- (base.right + (l === l))) by Restate
+                  val eqs = zip.map((l, r) => l === r)
+                  val goal = have((base.left ++ eqs) |- (base.right + (l === r))).by.bot
+                  have((base.left ++ eqs) |- (base.right + (l === r))) by RightSubstEq.withParametersSimple(zip, lambda(vars, l === labelr.applyUnsafe(children)))(lastStep)
+                  steps.foreach { s =>
+                    have(
+                      if s._2.bot.left.contains(s._1) then lastStep.bot else lastStep.bot -<< s._1
+                    ) by Cut(s._2, lastStep)
+                  }
+                case _ => 
+                  println(s"l: $l")
+                  println(s"r: $r")
+                  throw UnreachableException
+        
+            }
+            if id1 != l then
+              val goalSequent = base.left |- (base.right + (id1 === r))
+              val x = freshVariable(id1)
+              have(goalSequent +<< (l === r)) by RightSubstEq.withParametersSimple(List((l, r)), lambda(x, id1 === x))(prev)
+              have(goalSequent) by Cut(leqr, lastStep)
+        }
+    }
+  }
+
+  def proveFormula(using lib: Library, proof: lib.Proof)(id1: Formula, id2:Formula, base: Sequent): proof.ProofTacticJudgement = 
+    TacticSubproof { proveInnerFormula(id1, id2, base) }
+
+  def proveInnerFormula(using lib: Library, proof: lib.Proof)(id1: Formula, id2:Formula, base: Sequent): Unit = {
+    import lib.*
+    val steps = explain(id1, id2)
+    steps match {
+      case None => throw new Exception("No proof found in the egraph")
+      case Some(steps) => 
+        if steps.isEmpty then have(base.left |- (base.right + (id1 <=> id2))) by Restate
+        steps.foreach {
+          case FormulaExternal((l, r)) => 
+            val goalSequent = base.left |- (base.right + (id1 <=> r))
+            if l == id1 then 
+              have(goalSequent) by Restate
+            else
+              val x = freshVariableFormula(id1)
+              have(goalSequent) by RightSubstIff.withParametersSimple(List((l, r)), lambda(x, id1 <=> x))(lastStep)
+          case FormulaCongruence((l, r)) => 
+            val prev = if id1 != l then lastStep else null
+            val leqr = have(base.left |- (base.right + (l <=> r))) subproof { sp ?=>
+              (l, r) match
+                case (AppliedConnector(labell, argsl), AppliedConnector(labelr, argsr)) if labell == labelr && argsl.size == argsr.size => 
+                  var freshn = freshId((l.freeVariableFormulas ++ r.freeVariableFormulas).map(_.id), "n").no
+                  val ziped = (argsl zip argsr)
+                  var zip = List[(Formula, Formula)]()
+                  var children = List[Formula]()
+                  var vars = List[VariableFormula]()
+                  var steps = List[(Formula, sp.ProofStep)]()
+                  ziped.reverse.foreach { (al, ar) =>
+                    if al == ar then children = al :: children
+                    else {
+                      val x = VariableFormula(Identifier("n", freshn))
+                      freshn = freshn + 1
+                      children = x :: children
+                      vars = x :: vars
+                      steps = (al <=> ar, have(proveFormula(al, ar, base))) :: steps
+                      zip = (al, ar) :: zip
+                    }
+                  }
+                  have(base.left |- (base.right + (l <=> l))) by Restate
+                  val eqs = zip.map((l, r) => l <=> r)
+                  val goal = have((base.left ++ eqs) |- (base.right + (l <=> r))).by.bot
+                  have((base.left ++ eqs) |- (base.right + (l <=> r))) by RightSubstIff.withParametersSimple(zip, lambda(vars, l <=> labelr.applyUnsafe(children)))(lastStep)
+                  steps.foreach { s =>
+                    have(
+                      if s._2.bot.left.contains(s._1) then lastStep.bot else lastStep.bot -<< s._1
+                    ) by Cut(s._2, lastStep)
+                  }
+
+                case (AppliedPredicate(labell, argsl), AppliedPredicate(labelr, argsr)) if labell == labelr && argsl.size == argsr.size => 
+                  var freshn = freshId((l.freeVariableFormulas ++ r.freeVariableFormulas).map(_.id), "n").no
+                  val ziped = (argsl zip argsr)
+                  var zip = List[(Term, Term)]()
+                  var children = List[Term]()
+                  var vars = List[Variable]()
+                  var steps = List[(Formula, sp.ProofStep)]()
+                  ziped.reverse.foreach { (al, ar) =>
+                    if al == ar then children = al :: children
+                    else {
+                      val x = Variable(Identifier("n", freshn))
+                      freshn = freshn + 1
+                      children = x :: children
+                      vars = x :: vars
+                      steps = (al === ar, have(proveTerm(al, ar, base))) :: steps
+                      zip = (al, ar) :: zip
+                    }
+                  }
+                  have(base.left |- (base.right + (l <=> l))) by Restate
+                  val eqs = zip.map((l, r) => l === r)
+                  val goal = have((base.left ++ eqs) |- (base.right + (l <=> r))).by.bot
+                  have((base.left ++ eqs) |- (base.right + (l <=> r))) by RightSubstEq.withParametersSimple(zip, lambda(vars, l <=> labelr.applyUnsafe(children)))(lastStep)
+                  steps.foreach { s =>
+                    have(
+                      if s._2.bot.left.contains(s._1) then lastStep.bot else lastStep.bot -<< s._1
+                    ) by Cut(s._2, lastStep)
+                  }
+                case _ => 
+                  println(s"l: $l")
+                  println(s"r: $r")
+                  throw UnreachableException
+        
+            }
+            if id1 != l then
+              val goalSequent = base.left |- (base.right + (id1 <=> r))
+              val x = freshVariableFormula(id1)
+              have(goalSequent +<< (l <=> r)) by RightSubstIff.withParametersSimple(List((l, r)), lambda(x, id1 <=> x))(prev)
+              have(goalSequent) by Cut(leqr, lastStep)
+        
+        }
+    }
+  }
+
+
+}
\ No newline at end of file
diff --git a/lisa-sets/src/main/scala/lisa/maths/settheory/SetTheory2.scala b/lisa-sets/src/main/scala/lisa/maths/settheory/SetTheory2.scala
index b3d69dcc..f9599302 100644
--- a/lisa-sets/src/main/scala/lisa/maths/settheory/SetTheory2.scala
+++ b/lisa-sets/src/main/scala/lisa/maths/settheory/SetTheory2.scala
@@ -36,7 +36,7 @@ object SetTheory2 extends lisa.Main {
     have(thesis) by Tableau
   }
 
-  val functionalIsFunctional = Lemma(
+  val functionalIsFunctional = Theorem(
     ∀(x, in(x, A) ==> ∀(y, ∀(z, (P(x, y) /\ P(x, z)) ==> (y === z)))).substitute(P := lambda((A, B), Filter(A) /\ (B === Map(A)))) <=> top
   ) {
 
@@ -45,7 +45,7 @@ object SetTheory2 extends lisa.Main {
     thenHave(in(x, A) |- ((Filter(x) /\ (y === Map(x)) /\ (z === Map(x))) ==> (y === z))) by Weakening
     thenHave(in(x, A) |- ∀(z, ((Filter(x) /\ (y === Map(x)) /\ (z === Map(x))) ==> (y === z)))) by RightForall
     thenHave(in(x, A) |- ∀(y, ∀(z, ((Filter(x) /\ (y === Map(x)) /\ (z === Map(x))) ==> (y === z))))) by RightForall
-    thenHave(in(x, A) ==> ∀(y, ∀(z, ((Filter(x) /\ (y === Map(x)) /\ (z === Map(x))) ==> (y === z))))) by Restate
+    //thenHave(in(x, A) ==> ∀(y, ∀(z, ((Filter(x) /\ (y === Map(x)) /\ (z === Map(x))) ==> (y === z))))) by Restate
     thenHave(∀(x, in(x, A) ==> ∀(y, ∀(z, ((Filter(x) /\ (y === Map(x)) /\ (z === Map(x))) ==> (y === z)))))) by RightForall
     thenHave(thesis) by Restate
 
diff --git a/lisa-sets/src/test/scala/lisa/automation/CongruenceTest.scala b/lisa-sets/src/test/scala/lisa/automation/CongruenceTest.scala
new file mode 100644
index 00000000..5e50502d
--- /dev/null
+++ b/lisa-sets/src/test/scala/lisa/automation/CongruenceTest.scala
@@ -0,0 +1,914 @@
+package lisa.automation
+import lisa.fol.FOL.{*, given}
+import lisa.automation.Congruence.*
+import lisa.automation.Congruence
+import org.scalatest.funsuite.AnyFunSuite
+
+
+class CongruenceTest extends AnyFunSuite with lisa.TestMain {
+
+  given lib: lisa.SetTheoryLibrary.type = lisa.SetTheoryLibrary
+
+  val a = variable
+  val b = variable
+  val c = variable
+  val d = variable
+  val e = variable
+  val f = variable
+  val g = variable
+  val h = variable
+  val i = variable
+  val j = variable
+  val k = variable
+  val l = variable
+  val m = variable
+  val n = variable
+  val o = variable
+
+  val x = variable
+
+  val F = function[1]
+
+  val one = variable
+  val two = variable
+  val * = SchematicFunctionLabel("*", 2)
+  val << = SchematicFunctionLabel("<<", 2)
+  val / = SchematicFunctionLabel("/", 2)
+
+  
+  val af = formulaVariable
+  val bf = formulaVariable
+  val cf = formulaVariable
+  val df = formulaVariable
+  val ef = formulaVariable
+  val ff = formulaVariable
+  val gf = formulaVariable
+  val hf = formulaVariable
+  val if_ = formulaVariable
+  val jf = formulaVariable
+  val kf = formulaVariable
+  val lf = formulaVariable
+  val mf = formulaVariable
+  val nf = formulaVariable
+  val of = formulaVariable
+
+  val xf = formulaVariable
+
+  val Ff = SchematicConnectorLabel("Ff", 1)
+  val Fp = SchematicPredicateLabel("Fp", 1)
+
+  val onef = formulaVariable
+  val twof = formulaVariable
+  val `*f` = SchematicConnectorLabel("*f", 2)
+  val `<<f` = SchematicConnectorLabel("<<f", 2)
+  val `/f` = SchematicConnectorLabel("/f", 2)
+
+
+  test("3 terms no congruence egraph test") {
+    val egraph = new EGraphTerms()
+    egraph.add(a)
+    egraph.add(b)
+    egraph.add(c)
+    egraph.merge(a, b)
+    assert(egraph.idEq(a, b))
+    assert(!egraph.idEq(a, c))
+
+  }
+
+  test("8 terms no congruence egraph test") {
+    val egraph = new EGraphTerms()
+    egraph.add(a)
+    egraph.add(b)
+    egraph.add(c)
+    egraph.add(d)
+    egraph.add(e)
+    egraph.add(f)
+    egraph.add(g)
+    egraph.add(h)
+    egraph.merge(a, b)
+    egraph.merge(c, d)
+    egraph.merge(e, f)
+    egraph.merge(g, h)
+    egraph.merge(a, c)
+    egraph.merge(f, h)
+    egraph.merge(a, f)
+    assert(egraph.idEq(a, h))
+
+  }
+
+  test("15 terms no congruence egraph test") {
+    val egraph = new EGraphTerms()
+    egraph.add(a)
+    egraph.add(b)
+    egraph.add(c)
+    egraph.add(d)
+    egraph.add(e)
+    egraph.add(f)
+    egraph.add(g)
+    egraph.add(h)
+    egraph.add(i)
+    egraph.add(j)
+    egraph.add(k)
+    egraph.add(l)
+    egraph.add(m)
+    egraph.add(n)
+    egraph.add(o)
+    egraph.merge(a, c)
+    egraph.merge(e, f)
+    egraph.merge(i, k)
+    egraph.merge(m, n)
+    egraph.merge(a, b)
+    egraph.merge(o, m)
+    egraph.merge(i, m)
+    egraph.merge(g, h)
+    egraph.merge(l, k)
+    egraph.merge(c, d)
+    egraph.merge(a, e)
+    egraph.merge(a, i)
+    egraph.merge(g, e)
+    egraph.merge(i, j)
+    assert(egraph.idEq(a, o))
+
+  }
+
+  test("15 terms no congruence egraph test with redundant merges") {
+    val egraph = new EGraphTerms()
+    egraph.add(a)
+    egraph.add(b)
+    egraph.add(c)
+    egraph.add(d)
+    egraph.add(e)
+    egraph.add(f)
+    egraph.add(g)
+    egraph.add(h)
+    egraph.add(i)
+    egraph.add(j)
+    egraph.add(k)
+    egraph.add(l)
+    egraph.add(m)
+    egraph.add(n)
+    egraph.add(o)
+    egraph.merge(a, c)
+    egraph.merge(e, f)
+    egraph.merge(i, k)
+    egraph.merge(m, n)
+    egraph.merge(a, b)
+    egraph.merge(o, m)
+    egraph.merge(i, m)
+    egraph.merge(g, h)
+    egraph.merge(l, k)
+    egraph.merge(b, c)
+    egraph.merge(f, e)
+    egraph.merge(o, i)
+    egraph.merge(g, e)
+    egraph.merge(i, j)
+
+    assert(egraph.idEq(b, c))
+    assert(egraph.idEq(f, h))
+    assert(egraph.idEq(i, o))
+    assert(!egraph.idEq(a, d))
+    assert(!egraph.idEq(b, g))
+    assert(!egraph.idEq(f, i))
+    assert(!egraph.idEq(n, c))
+    assert(egraph.termUF.getClasses.size == 4)
+
+  }
+
+  test("4 terms withcongruence egraph test") {
+    val egraph = new EGraphTerms()
+    egraph.add(F(a))
+    egraph.add(F(b))
+    egraph.merge(a, b)
+    assert(egraph.idEq(a, b))
+    assert(egraph.idEq(F(a), F(b)))
+    assert(!egraph.idEq(a, F(a)))
+    assert(!egraph.idEq(a, F(b)))
+    assert(!egraph.idEq(b, F(a)))
+    assert(!egraph.idEq(b, F(b)))
+
+    assert(egraph.explain(F(a), F(b)) == Some(List(egraph.TermCongruence((F(a), F(b))))) )
+
+  }
+
+
+
+  test("divide-mult-shift in terms by 2 egraph test") {
+
+    val egraph = new EGraphTerms()
+    egraph.add(one)
+    egraph.add(two)
+    egraph.add(a)
+    val ax2    = egraph.add(*(a, two))
+    val ax2_d2 = egraph.add(/(*(a, two), two))
+    val `2d2`  = egraph.add(/(two, two))
+    val ax_2d2 = egraph.add(*(a, /(two, two)))
+    val ax1    = egraph.add(*(a, one))
+    val as1    = egraph.add(<<(a, one))
+
+    egraph.merge(ax2, as1)
+    egraph.merge(ax2_d2, ax_2d2)
+    egraph.merge(`2d2`, one)
+    egraph.merge(ax1, a)
+
+
+    assert(egraph.idEq(one, `2d2`))
+    assert(egraph.idEq(ax2, as1))
+    assert(egraph.idEq(ax2_d2, ax_2d2))
+    assert(egraph.idEq(ax_2d2, ax1))
+    assert(egraph.idEq(ax_2d2, a))
+
+    assert(!egraph.idEq(ax2, ax2_d2))
+    assert(!egraph.idEq(ax2, `2d2`))
+    assert(!egraph.idEq(ax2, ax_2d2))
+    assert(!egraph.idEq(ax2, ax1))
+    assert(!egraph.idEq(ax2, a))
+    assert(!egraph.idEq(ax2_d2, `2d2`))
+
+    assert(egraph.explain(one, `2d2`) == Some(List(egraph.TermExternal((one, `2d2`)))) )
+    assert(egraph.explain(ax2, as1) == Some(List(egraph.TermExternal((ax2, as1)))) )
+    assert(egraph.explain(ax2_d2, ax_2d2) == Some(List(egraph.TermExternal((ax2_d2, ax_2d2)))) )
+
+    assert(egraph.explain(ax_2d2, ax1) == Some(List(egraph.TermCongruence((ax_2d2, ax1)))) )
+    assert(egraph.explain(ax_2d2, a) == Some(List(egraph.TermCongruence((ax_2d2, ax1)), egraph.TermExternal((ax1, a))) ))
+
+
+  }
+
+  test("long chain of terms congruence eGraph") {
+    val egraph = new EGraphTerms()
+    egraph.add(x)
+    val fx = egraph.add(F(x))
+    val ffx = egraph.add(F(fx))
+    val fffx = egraph.add(F(ffx))
+    val ffffx = egraph.add(F(fffx))
+    val fffffx = egraph.add(F(ffffx))
+    val ffffffx = egraph.add(F(fffffx))
+    val fffffffx = egraph.add(F(ffffffx))
+    val ffffffffx = egraph.add(F(fffffffx))
+
+
+    egraph.merge(ffffffffx, x)
+    egraph.merge(fffffx, x)
+    assert(egraph.idEq(fffx, x))
+    assert(egraph.idEq(ffx, x))
+    assert(egraph.idEq(fx, x))
+    assert(egraph.idEq(x, fx))
+
+    assert(egraph.explain(fx, x) == Some(List(egraph.TermCongruence(fx, fffx), egraph.TermCongruence(fffx, ffffffffx), egraph.TermExternal(ffffffffx, x))))
+
+  }
+
+
+  test("3 formulas no congruence egraph test") {
+    val egraph = new EGraphTerms()
+    egraph.add(af)
+    egraph.add(bf)
+    egraph.add(cf)
+    egraph.merge(af, bf)
+    assert(egraph.idEq(af, bf))
+    assert(!egraph.idEq(af, cf))
+
+  }
+
+  test("8 formulas no congruence egraph test") {
+    val egraph = new EGraphTerms()
+    egraph.add(af)
+    egraph.add(bf)
+    egraph.add(cf)
+    egraph.add(df)
+    egraph.add(ef)
+    egraph.add(ff)
+    egraph.add(gf)
+    egraph.add(hf)
+    egraph.merge(af, bf)
+    egraph.merge(cf, df)
+    egraph.merge(ef, ff)
+    egraph.merge(gf, hf)
+    egraph.merge(af, cf)
+    egraph.merge(ff, hf)
+    egraph.merge(af, ff)
+    assert(egraph.idEq(af, hf))
+
+  }
+
+  test("15 formulas no congruence egraph test") {
+    val egraph = new EGraphTerms()
+    egraph.add(af)
+    egraph.add(bf)
+    egraph.add(cf)
+    egraph.add(df)
+    egraph.add(ef)
+    egraph.add(ff)
+    egraph.add(gf)
+    egraph.add(hf)
+    egraph.add(if_)
+    egraph.add(jf)
+    egraph.add(kf)
+    egraph.add(lf)
+    egraph.add(mf)
+    egraph.add(nf)
+    egraph.add(of)
+    egraph.merge(af, cf)
+    egraph.merge(ef, ff)
+    egraph.merge(if_, kf)
+    egraph.merge(mf, nf)
+    egraph.merge(af, bf)
+    egraph.merge(of, mf)
+    egraph.merge(if_, mf)
+    egraph.merge(gf, hf)
+    egraph.merge(lf, kf)
+    egraph.merge(cf, df)
+    egraph.merge(af, ef)
+    egraph.merge(af, if_)
+    egraph.merge(gf, ef)
+    egraph.merge(if_, jf)
+    assert(egraph.idEq(af, of))
+
+  }
+
+  test("15 formulas no congruence egraph test with redundant merges") {
+    val egraph = new EGraphTerms()
+    egraph.add(af)
+    egraph.add(bf)
+    egraph.add(cf)
+    egraph.add(df)
+    egraph.add(ef)
+    egraph.add(ff)
+    egraph.add(gf)
+    egraph.add(hf)
+    egraph.add(if_)
+    egraph.add(jf)
+    egraph.add(kf)
+    egraph.add(lf)
+    egraph.add(mf)
+    egraph.add(nf)
+    egraph.add(of)
+    egraph.merge(af, cf)
+    egraph.merge(ef, ff)
+    egraph.merge(if_, kf)
+    egraph.merge(mf, nf)
+    egraph.merge(af, bf)
+    egraph.merge(of, mf)
+    egraph.merge(if_, mf)
+    egraph.merge(gf, hf)
+    egraph.merge(lf, kf)
+    egraph.merge(bf, cf)
+    egraph.merge(ff, ef)
+    egraph.merge(of, if_)
+    egraph.merge(gf, ef)
+    egraph.merge(if_, jf)
+
+    assert(egraph.idEq(bf, cf))
+    assert(egraph.idEq(ff, hf))
+    assert(egraph.idEq(if_, of))
+    assert(!egraph.idEq(af, df))
+    assert(!egraph.idEq(bf, gf))
+    assert(!egraph.idEq(ff, if_))
+    assert(!egraph.idEq(nf, cf))
+    assert(egraph.formulaUF.getClasses.size == 4)
+
+  }
+
+  test("4 formulas withcongruence egraph test") {
+    val egraph = new EGraphTerms()
+    egraph.add(Ff(af))
+    egraph.add(Ff(bf))
+    egraph.merge(af, bf)
+    assert(egraph.idEq(af, bf))
+    assert(egraph.idEq(Ff(af), Ff(bf)))
+    assert(!egraph.idEq(af, Ff(af)))
+    assert(!egraph.idEq(af, Ff(bf)))
+    assert(!egraph.idEq(bf, Ff(af)))
+    assert(!egraph.idEq(bf, Ff(bf)))
+
+    assert(egraph.explain(Ff(af), Ff(bf)) == Some(List(egraph.FormulaCongruence((Ff(af), Ff(bf))))) )
+
+  }
+
+  test("divide-mult-shift in formulas by 2 egraph test") {
+
+    val egraph = new EGraphTerms()
+    egraph.add(onef)
+    egraph.add(twof)
+    egraph.add(af)
+    val ax2    = egraph.add(`*f`(af, twof))
+    val ax2_d2 = egraph.add(`/f`(`*f`(af, twof), twof))
+    val `2d2`  = egraph.add(`/f`(twof, twof))
+    val ax_2d2 = egraph.add(`*f`(af, `/f`(twof, twof)))
+    val ax1    = egraph.add(`*f`(af, onef))
+    val as1    = egraph.add(`<<f`(af, onef))
+
+    egraph.merge(ax2, as1)
+    egraph.merge(ax2_d2, ax_2d2)
+    egraph.merge(`2d2`, onef)
+    egraph.merge(ax1, af)
+
+
+    assert(egraph.idEq(onef, `2d2`))
+    assert(egraph.idEq(ax2, as1))
+    assert(egraph.idEq(ax2_d2, ax_2d2))
+    assert(egraph.idEq(ax_2d2, ax1))
+    assert(egraph.idEq(ax_2d2, af))
+
+    assert(!egraph.idEq(ax2, ax2_d2))
+    assert(!egraph.idEq(ax2, `2d2`))
+    assert(!egraph.idEq(ax2, ax_2d2))
+    assert(!egraph.idEq(ax2, ax1))
+    assert(!egraph.idEq(ax2, af))
+    assert(!egraph.idEq(ax2_d2, `2d2`))
+
+    assert(egraph.explain(onef, `2d2`) == Some(List(egraph.FormulaExternal((onef, `2d2`)))) )
+    assert(egraph.explain(ax2, as1) == Some(List(egraph.FormulaExternal((ax2, as1)))) )
+    assert(egraph.explain(ax2_d2, ax_2d2) == Some(List(egraph.FormulaExternal((ax2_d2, ax_2d2)))) )
+
+    assert(egraph.explain(ax_2d2, ax1) == Some(List(egraph.FormulaCongruence((ax_2d2, ax1)))) )
+    assert(egraph.explain(ax_2d2, af) == Some(List(egraph.FormulaCongruence((ax_2d2, ax1)), egraph.FormulaExternal((ax1, af))) ))
+
+    
+
+
+  }
+
+  test("long chain of formulas congruence eGraph") {
+    val egraph = new EGraphTerms()
+    egraph.add(xf)
+    val fx = egraph.add(Ff(xf))
+    val ffx = egraph.add(Ff(fx))
+    val fffx = egraph.add(Ff(ffx))
+    val ffffx = egraph.add(Ff(fffx))
+    val fffffx = egraph.add(Ff(ffffx))
+    val ffffffx = egraph.add(Ff(fffffx))
+    val fffffffx = egraph.add(Ff(ffffffx))
+    val ffffffffx = egraph.add(Ff(fffffffx))
+
+
+    egraph.merge(ffffffffx, xf)
+    egraph.merge(fffffx, xf)
+    assert(egraph.idEq(fffx, xf))
+    assert(egraph.idEq(ffx, xf))
+    assert(egraph.idEq(fx, xf))
+    assert(egraph.idEq(xf, fx))
+    assert(egraph.explain(fx, xf) == Some(List(egraph.FormulaCongruence(fx, ffffffffx), egraph.FormulaExternal(ffffffffx, xf))))
+
+  }
+
+  //////////////////////////////////////
+  //// With both terms and formulas ////
+  //////////////////////////////////////
+
+  test("2 terms 6 predicates with congruence egraph test") {
+    val egraph = new EGraphTerms()
+    egraph.add(Ff(Ff(Fp(a))))
+    egraph.add(Ff(Ff(Fp(b))))
+    egraph.merge(a, b)
+    assert(egraph.idEq(a, b))
+    assert(egraph.idEq(Fp(a), Fp(b)))
+    assert(egraph.idEq(Ff(Fp(a)), Ff(Fp(b))))
+    assert(egraph.idEq(Ff(Ff(Fp(a))), Ff(Ff(Fp(b)))))
+    assert(!egraph.idEq(Fp(a), Ff(Fp(a))))
+    assert(!egraph.idEq(Fp(a), Ff(Fp(b))))
+    assert(!egraph.idEq(Fp(b), Ff(Fp(a))))
+    assert(!egraph.idEq(Fp(b), Ff(Ff(Fp(b)))))
+    assert(!egraph.idEq(Ff(Fp(a)), Ff(Ff(Fp(b)))))
+    assert(egraph.formulaUF.getClasses.size == 3)
+
+    egraph.merge(Fp(a), Ff(Fp(a)))
+    assert(egraph.idEq(Fp(a), Ff(Fp(b))))
+    assert(egraph.idEq(Fp(b), Ff(Fp(a))))
+    assert(egraph.idEq(Ff(Fp(a)), Ff(Ff(Fp(a)))))
+    assert(egraph.idEq(Fp(b), Ff(Ff(Fp(a)))))
+    assert(egraph.formulaUF.getClasses.size == 1)
+
+  }
+
+    test("6 terms 6 predicates with congruence egraph test") {
+    val egraph = new EGraphTerms()
+    egraph.add(Ff(Ff(Fp(F(F(a))))))
+    egraph.add(Ff(Ff(Fp(F(F(b))))))
+    egraph.merge(a, b)
+    assert(egraph.idEq(a, b))
+    assert(egraph.idEq(F(a), F(b)))
+    assert(egraph.idEq(Fp(F(F(a))), Fp(F(F(b)))))
+    assert(egraph.idEq(Ff(Ff(Fp(F(F(a))))), Ff(Ff(Fp(F(F(b)))))))
+    assert(egraph.formulaUF.getClasses.size == 3)
+    assert(egraph.termUF.getClasses.size == 3)
+
+    egraph.merge(Fp(F(F(b))), Ff(Fp(F(F(a)))))
+    assert(egraph.formulaUF.getClasses.size == 1)
+
+  }
+
+
+  test("15 terms no congruence with redundant merges test with proofs") {
+    val egraph = new EGraphTerms()
+    egraph.add(a)
+    egraph.add(b)
+    egraph.add(c)
+    egraph.add(d)
+    egraph.add(e)
+    egraph.add(f)
+    egraph.add(g)
+    egraph.add(h)
+    egraph.add(i)
+    egraph.add(j)
+    egraph.add(k)
+    egraph.add(l)
+    egraph.add(m)
+    egraph.add(n)
+    egraph.add(o)
+    egraph.merge(a, c)
+    egraph.merge(e, f)
+    egraph.merge(i, k)
+    egraph.merge(m, n)
+    egraph.merge(a, b)
+    egraph.merge(o, m)
+    egraph.merge(i, m)
+    egraph.merge(g, h)
+    egraph.merge(l, k)
+    egraph.merge(b, c)
+    egraph.merge(f, e)
+    egraph.merge(o, i)
+    egraph.merge(g, e)
+    egraph.merge(i, j)
+    val base = List(a === c, e === f, i === k, m === n, a === b, o === m, i === m, g === h, l === k, b === c, f === e, o === i, g === e, i === j)
+
+    val test1 = Theorem(base |- (b === c)) {
+      egraph.proveInnerTerm(b, c, base |- ())
+    }
+
+    val test2 = Theorem(base |- (f === h)) {
+      egraph.proveInnerTerm(f, h, base |- ())
+    }
+
+    val test3 = Theorem(base |- (i === o)) {
+      egraph.proveInnerTerm(i, o, base |- ())
+    }
+
+    val test4 = Theorem(base |- (o === i)) {
+      egraph.proveInnerTerm(o, i, base |- ())
+    }
+
+  }
+
+
+  test("4 elements with congruence test with proofs") {
+    val egraph = new EGraphTerms()
+    egraph.add(F(a))
+    egraph.add(F(b))
+    egraph.merge(a, b)
+    val test5 = Theorem(a===b |- F(a) === F(b)) {
+      egraph.proveInnerTerm(F(a), F(b), (a === b) |- ())
+    }
+  }
+
+
+  test("divide-mult-shift by 2 in terms egraph test with proofs") {
+    val egraph = new EGraphTerms()
+    egraph.add(one)
+    egraph.add(two)
+    egraph.add(a)
+    val ax2    = egraph.add(`*`(a, two))
+    val ax2_d2 = egraph.add(`/`(`*`(a, two), two))
+    val `2d2`  = egraph.add(`/`(two, two))
+    val ax_2d2 = egraph.add(`*`(a, `/`(two, two)))
+    val ax1    = egraph.add(`*`(a, one))
+    val as1    = egraph.add(`<<`(a, one))
+
+    egraph.merge(ax2, as1)
+    egraph.merge(ax2_d2, ax_2d2)
+    egraph.merge(`2d2`, one)
+    egraph.merge(ax1, a)
+
+    val base = List[Formula](ax2 === as1, ax2_d2 === ax_2d2, `2d2` === one, ax1 === a)
+
+    val one_2d2 = Theorem(base |- (one === `2d2`)) {
+      egraph.proveInnerTerm(one, `2d2`, base  |- ())
+    }
+
+    val ax2_as1 = Theorem(base |- (ax2 === as1)) {
+      egraph.proveInnerTerm(ax2, as1, base  |- ())
+    }
+
+    val ax2_d2_ax_2d2 = Theorem(base |- (ax2_d2 === ax_2d2)) {
+      egraph.proveInnerTerm(ax2_d2, ax_2d2, base  |- ())
+    }
+
+    val ax_2d2_ax1 = Theorem(base |- (ax_2d2 === ax1)) {
+      egraph.proveInnerTerm(ax_2d2, ax1, base  |- ())
+    }
+
+    val ax_2d2_a = Theorem(base |- (ax_2d2 === a)) {
+      egraph.proveInnerTerm(ax_2d2, a, base  |- ())
+    }
+
+  }
+
+  test("long chain of termscongruence eGraph with proofs") {
+    val egraph = new EGraphTerms()
+    egraph.add(x)
+    val fx = egraph.add(F(x))
+    val ffx = egraph.add(F(fx))
+    val fffx = egraph.add(F(ffx))
+    val ffffx = egraph.add(F(fffx))
+    val fffffx = egraph.add(F(ffffx))
+    val ffffffx = egraph.add(F(fffffx))
+    val fffffffx = egraph.add(F(ffffffx))
+    val ffffffffx = egraph.add(F(fffffffx))
+
+    egraph.merge(ffffffffx, x)
+    egraph.merge(fffffx, x)
+
+
+    val base = List(ffffffffx === x, fffffx === x)
+
+  
+    val test2 = Theorem(base |- fffx === x) {
+      egraph.proveInnerTerm(fffx, x, base |- ())
+    }
+    val test3 = Theorem(base |- ffx === x) {
+      egraph.proveInnerTerm(ffx, x, base |- ())
+    }
+    val test4 = Theorem(base |- fx === x) {
+      egraph.proveInnerTerm(fx, x, base |- ())
+    }
+
+  }
+
+
+  test("15 formulas no congruence proofs with redundant merges test with proofs") {
+    val egraph = new EGraphTerms()
+    egraph.add(af)
+    egraph.add(bf)
+    egraph.add(cf)
+    egraph.add(df)
+    egraph.add(ef)
+    egraph.add(ff)
+    egraph.add(gf)
+    egraph.add(hf)
+    egraph.add(if_)
+    egraph.add(jf)
+    egraph.add(kf)
+    egraph.add(lf)
+    egraph.add(mf)
+    egraph.add(nf)
+    egraph.add(of)
+    egraph.merge(af, cf)
+    egraph.merge(ef, ff)
+    egraph.merge(if_, kf)
+    egraph.merge(mf, nf)
+    egraph.merge(af, bf)
+    egraph.merge(of, mf)
+    egraph.merge(if_, mf)
+    egraph.merge(gf, hf)
+    egraph.merge(lf, kf)
+    egraph.merge(bf, cf)
+    egraph.merge(ff, ef)
+    egraph.merge(of, if_)
+    egraph.merge(gf, ef)
+    egraph.merge(if_, jf)
+
+    val base = List(af <=> cf, ef <=> ff, if_ <=> kf, mf <=> nf, af <=> bf,
+     of <=> mf, if_ <=> mf, gf <=> hf, lf <=> kf, bf <=> cf, ff <=> ef, of <=> if_, gf <=> ef, if_ <=> jf)
+
+    val test1 = Theorem(base |- bf <=> cf) {
+      egraph.proveInnerFormula(bf, cf, base |- ())
+    }
+
+    val test2 = Theorem(base |- ff <=> hf) {
+      egraph.proveInnerFormula(ff, hf, base |- ())
+    }
+
+    val test3 = Theorem(base |- if_ <=> of) {
+      egraph.proveInnerFormula(if_, of, base |- ())
+    }
+
+    val test4 = Theorem(base |- of <=> if_) {
+      egraph.proveInnerFormula(of, if_, base |- ())
+    }
+
+  }
+
+  test("4 formulas with congruence test with proofs") {
+    val egraph = new EGraphTerms()
+    egraph.add(Ff(af))
+    egraph.add(Ff(bf))
+    egraph.merge(af, bf)
+    val test5 = Theorem(af <=> bf |- Ff(af) <=> Ff(bf)) {
+      egraph.proveInnerFormula(Ff(af), Ff(bf), (af <=> bf) |- ())
+    }
+  }
+
+  test("divide-mult-shift by 2 in formulas egraph test with proofs") {
+    val egraph = new EGraphTerms()
+    egraph.add(onef)
+    egraph.add(twof)
+    egraph.add(af)
+    val ax2    = egraph.add(`*f`(af, twof))
+    val ax2_d2 = egraph.add(`/f`(`*f`(af, twof), twof))
+    val `2d2`  = egraph.add(`/f`(twof, twof))
+    val ax_2d2 = egraph.add(`*f`(af, `/f`(twof, twof)))
+    val ax1    = egraph.add(`*f`(af, onef))
+    val as1    = egraph.add(`<<f`(af, onef))
+
+    egraph.merge(ax2, as1)
+    egraph.merge(ax2_d2, ax_2d2)
+    egraph.merge(`2d2`, onef)
+    egraph.merge(ax1, af)
+
+    val base = List[Formula](ax2 <=> as1, ax2_d2 <=> ax_2d2, `2d2` <=> onef, ax1 <=> af)
+
+    val one_2d2 = Theorem(base |- onef <=> `2d2`) {
+      egraph.proveInnerFormula(onef, `2d2`, base  |- ())
+    }
+
+    val ax2_as1 = Theorem(base |- ax2 <=> as1) {
+      egraph.proveInnerFormula(ax2, as1, base  |- ())
+    }
+
+    val ax2_d2_ax_2d2 = Theorem(base |- ax2_d2 <=> ax_2d2) {
+      egraph.proveInnerFormula(ax2_d2, ax_2d2, base  |- ())
+    }
+
+    val ax_2d2_ax1 = Theorem(base |- ax_2d2 <=> ax1) {
+      egraph.proveInnerFormula(ax_2d2, ax1, base  |- ())
+    }
+
+    val ax_2d2_a = Theorem(base |- ax_2d2 <=> af) {
+      egraph.proveInnerFormula(ax_2d2, af, base  |- ())
+    }
+
+  }
+
+  test("long chain of formulas congruence eGraph with proofs") {
+    val egraph = new EGraphTerms()
+    egraph.add(xf)
+    val fx = egraph.add(Ff(xf))
+    val ffx = egraph.add(Ff(fx))
+    val fffx = egraph.add(Ff(ffx))
+    val ffffx = egraph.add(Ff(fffx))
+    val fffffx = egraph.add(Ff(ffffx))
+    val ffffffx = egraph.add(Ff(fffffx))
+    val fffffffx = egraph.add(Ff(ffffffx))
+    val ffffffffx = egraph.add(Ff(fffffffx))
+
+    egraph.merge(ffffffffx, xf)
+    egraph.merge(fffffx, xf)
+    
+    val base = List(ffffffffx <=> xf, fffffx <=> xf)
+
+    val test2 = Theorem(base |- fffx <=> xf) {
+      egraph.proveInnerFormula(fffx, xf, base |- ())
+    }
+    val test3 = Theorem(base |- ffx <=> xf) {
+      egraph.proveInnerFormula(ffx, xf, base |- ())
+    }
+    val test4 = Theorem(base |- fx <=> xf) {
+      egraph.proveInnerFormula(fx, xf, base |- ())
+    }
+  }
+
+
+  test("2 terms 6 predicates with congruence egraph test with proofs") {
+    val egraph = new EGraphTerms()
+    egraph.add(Ff(Ff(Fp(a))))
+    egraph.add(Ff(Ff(Fp(b))))
+    egraph.merge(a, b)
+
+    val test5 = Theorem((a === b) |- Fp(a) <=> Fp(b)) {
+      egraph.proveInnerFormula(Fp(a), Fp(b), (a === b) |- ())
+    }
+
+    val test6 = Theorem((a === b) |- Ff(Fp(a)) <=> Ff(Fp(b))) {
+      egraph.proveInnerFormula(Ff(Fp(a)), Ff(Fp(b)), (a === b) |- ())
+    }
+
+    val test7 = Theorem((a === b) |- Ff(Ff(Fp(a))) <=> Ff(Ff(Fp(b))) ) {
+      egraph.proveInnerFormula(Ff(Ff(Fp(a))), Ff(Ff(Fp(b))), (a === b) |- ())
+    }
+
+  }
+
+  test("6 terms 6 predicates with congruence egraph test with proofs") {
+    val egraph = new EGraphTerms()
+    egraph.add(Ff(Ff(Fp(F(F(a))))))
+    egraph.add(Ff(Ff(Fp(F(F(b))))))
+    egraph.merge(a, b)
+
+    val test5 = Theorem((a === b) |- (F(a) === F(b))) {
+      egraph.proveInnerTerm(F(a), F(b), (a === b) |- ())
+    }
+
+    val test6 = Theorem((a === b) |- Fp(F(F(a))) <=> Fp(F(F(b))) ) {
+      egraph.proveInnerFormula(Fp(F(F(a))), Fp(F(F(b))), (a === b) |- ())
+    }
+
+    val test7 = Theorem((a === b) |- Ff(Ff(Fp(F(F(a))))) <=> Ff(Ff(Fp(F(F(b))))) ) {
+      egraph.proveInnerFormula(Ff(Ff(Fp(F(F(a))))), Ff(Ff(Fp(F(F(b))))), (a === b) |- ())
+    }
+
+    egraph.merge(Fp(F(F(b))), Ff(Fp(F(F(a)))))
+
+    val test8 = Theorem(((a === b), Fp(F(F(b))) <=> Ff(Fp(F(F(a)))) ) |- Ff(Ff(Fp(F(F(a))))) <=> Ff(Ff(Fp(F(F(b))))) ) {
+      egraph.proveInnerFormula(Ff(Ff(Fp(F(F(a))))), Ff(Ff(Fp(F(F(b))))), (a === b, Fp(F(F(b))) <=> Ff(Fp(F(F(a)))) ) |- ())
+    }
+
+  }
+
+  test("Full congruence tactic tests") {
+    println("Full congruence tactic tests\n")
+
+    val base1 = List(a === c, e === f, i === k, m === n, a === b, o === m, i === m, g === h, l === k, b === c, f === e, o === i, g === e, i === j)
+
+    val test1 = Theorem(base1 |- (b === c)) {
+      have(thesis) by Congruence
+    }
+
+    val test2 = Theorem(base1 |- (f === h)) {
+      have(thesis) by Congruence
+    }
+
+    val test3 = Theorem(base1 |- (i === o)) {
+      have(thesis) by Congruence
+    }
+
+
+    val ax2    = `*`(a, two)
+    val ax2_d2 = `/`(`*`(a, two), two)
+    val `2d2`  = `/`(two, two)
+    val ax_2d2 = `*`(a, `/`(two, two))
+    val ax1    = `*`(a, one)
+    val as1    = `<<`(a, one)
+
+    val base2 = List[Formula](ax2 === as1, ax2_d2 === ax_2d2, `2d2` === one, ax1 === a)
+
+
+    val one_2d2 = Theorem(base2 |- (one === `2d2`)) {
+      have(thesis) by Congruence
+    }
+
+    val ax2_as1 = Theorem(base2 |- (ax2 === as1)) {
+      have(thesis) by Congruence
+    }
+
+    val ax2_d2_ax_2d2 = Theorem(base2 |- (ax2_d2 === ax_2d2)) {
+      have(thesis) by Congruence
+    }
+
+    val ax_2d2_ax1 = Theorem(base2 |- (ax_2d2 === ax1)) {
+      have(thesis) by Congruence
+    }
+    
+    val ax_2d2_a = Theorem(base2 |- (ax_2d2 === a)) {
+      have(thesis) by Congruence
+    }
+
+    val ax_2d2_a_2 = Theorem(base2 |- (Fp(ax_2d2) <=> Fp(a))) {
+      have(thesis) by Congruence
+    }
+
+    val ax_2d2_a_1 = Theorem((Fp(a) :: base2) |- Fp(ax_2d2)) {
+      have(thesis) by Congruence
+    }
+
+    val ax_2d2_a_3 = Theorem((base2 :+ Fp(ax_2d2) :+ !Fp(a)) |- () ) {
+      have(thesis) by Congruence
+    }
+
+    val test5 = Theorem(a===b |- F(a) === F(b)) {
+      have(thesis) by Congruence
+    }
+
+    val test6 = Theorem(a === b |- F(a) === F(b)) {
+      have(thesis) by Congruence
+    }
+
+    val test7 = Theorem((Ff(Ff(Ff(Ff(Ff(Ff(Ff(xf))))))) <=> xf, Ff(Ff(Ff(Ff(Ff(xf))))) <=> xf) |- Ff(Ff(Ff(xf))) <=> xf) {
+      have(thesis) by Congruence
+    }
+
+    val test8 = Theorem((Ff(Ff(Ff(Ff(Ff(Ff(Ff(xf))))))) <=> xf, Ff(Ff(Ff(Ff(Ff(xf))))) <=> xf) |- Ff(xf) <=> xf) {
+      have(thesis) by Congruence
+    }
+
+    val test9 = Theorem((a === b) |- (Fp(F(F(a))), !Fp(F(F(b)))) ) {
+      have(thesis) by Congruence
+    }
+
+    val test10 = Theorem((a === b) |- Fp(F(F(a))) <=> Fp(F(F(b))) ) {
+      have(thesis) by Congruence
+    }
+
+    
+    val test11 = Theorem((a === b) |- Ff(Ff(Fp(F(F(a))))) <=> Ff(Ff(Fp(F(F(b))))) ) {
+      have(thesis) by Congruence
+    }
+
+    val test12 = Theorem(((a === b), Fp(F(F(b))) <=> Ff(Fp(F(F(a)))), Ff(Ff(Fp(F(F(a))))) ) |- Ff(Ff(Fp(F(F(b))))) ) {
+      have(thesis) by Congruence
+    }
+
+
+  }
+
+
+}
\ No newline at end of file
diff --git a/lisa-sets/src/test/scala/lisa/utilities/TestMain.scala b/lisa-sets/src/test/scala/lisa/utilities/TestMain.scala
new file mode 100644
index 00000000..6d93e5d0
--- /dev/null
+++ b/lisa-sets/src/test/scala/lisa/utilities/TestMain.scala
@@ -0,0 +1,16 @@
+package lisa
+
+import lisa.prooflib.*
+
+trait TestMain extends lisa.Main {
+
+  override val om: OutputManager = new OutputManager {
+    def finishOutput(exception: Exception): Nothing = {
+      log(exception)
+      main(Array[String]())
+      throw exception
+    }
+    val stringWriter: java.io.StringWriter = new java.io.StringWriter()
+  }
+
+}
diff --git a/lisa-utils/src/main/scala/lisa/fol/FOLHelpers.scala b/lisa-utils/src/main/scala/lisa/fol/FOLHelpers.scala
index 402ad81b..d9232d3b 100644
--- a/lisa-utils/src/main/scala/lisa/fol/FOLHelpers.scala
+++ b/lisa-utils/src/main/scala/lisa/fol/FOLHelpers.scala
@@ -50,6 +50,9 @@ object FOLHelpers {
   def predicate[N <: Arity: ValueOf](using name: sourcecode.Name): SchematicPredicateLabel[N] = SchematicPredicateLabel[N](name.value, valueOf[N])
   def connector[N <: Arity: ValueOf](using name: sourcecode.Name): SchematicConnectorLabel[N] = SchematicConnectorLabel[N](name.value, valueOf[N])
 
+  def freshVariable(using name: sourcecode.Name)(elems: LisaObject[?]*): Variable = Variable(freshId(elems.flatMap(_.freeVariables).map(_.id), name.value))
+  def freshVariableFormula(using name: sourcecode.Name)(elems: LisaObject[?]*): VariableFormula = VariableFormula(freshId(elems.flatMap(_.freeVariables).map(_.id), name.value))
+
   ////////////////////////////////////////
   //    Kernel to Front transformers    //
   ////////////////////////////////////////
diff --git a/lisa-utils/src/main/scala/lisa/prooflib/BasicMain.scala b/lisa-utils/src/main/scala/lisa/prooflib/BasicMain.scala
index 34f8878f..748f58d5 100644
--- a/lisa-utils/src/main/scala/lisa/prooflib/BasicMain.scala
+++ b/lisa-utils/src/main/scala/lisa/prooflib/BasicMain.scala
@@ -7,7 +7,7 @@ trait BasicMain {
 
   private val realOutput: String => Unit = println
 
-  given om: OutputManager = new OutputManager {
+  val om: OutputManager = new OutputManager {
     def finishOutput(exception: Exception): Nothing = {
       log(exception)
       main(Array[String]())
@@ -24,4 +24,6 @@ trait BasicMain {
     realOutput(om.stringWriter.toString)
   }
 
+  given om.type = om
+
 }
diff --git a/lisa-utils/src/main/scala/lisa/prooflib/ProofsHelpers.scala b/lisa-utils/src/main/scala/lisa/prooflib/ProofsHelpers.scala
index adda8986..946a16f2 100644
--- a/lisa-utils/src/main/scala/lisa/prooflib/ProofsHelpers.scala
+++ b/lisa-utils/src/main/scala/lisa/prooflib/ProofsHelpers.scala
@@ -21,8 +21,8 @@ trait ProofsHelpers {
 
   given Library = library
 
-  class HaveSequent /*private[ProofsHelpers]*/ (val bot: Sequent) {
-    // val x: lisa.fol.FOL.Sequent = bot
+  class HaveSequent(val bot: Sequent) {
+
     inline infix def by(using proof: library.Proof, line: sourcecode.Line, file: sourcecode.File): By { val _proof: proof.type } = By(proof, line, file).asInstanceOf
 
     class By(val _proof: library.Proof, line: sourcecode.Line, file: sourcecode.File) {
diff --git a/refman/lisa.pdf b/refman/lisa.pdf
index 80036345874145be30b7dc8a7edc9bd947f413f0..d22d7da5a928e5881571fad1be52a1ad76fd4a2c 100644
GIT binary patch
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diff --git a/refman/tactics.tex b/refman/tactics.tex
index d3c92691..c7030feb 100644
--- a/refman/tactics.tex
+++ b/refman/tactics.tex
@@ -1,6 +1,59 @@
 \chapter{Tactics: Specifications and Use}
 \label{chapt:tactics}
 
+\subsection*{Congruence}
+The \lstinline|Congruence| tactic is used to prove sequents whose validity directly follow from the congruence closure of all equalities and formula equivalences given left of the sequent.
+Specifically, it works in the following cases:
+\begin{itemize}
+  \item The right side contains an equality s === t or equivalence a <=> b provable in the congruence closure.
+  \item The left side contains an negated equality !(s === t) or equivalence !(a <=> b) provable in the congruence closure.
+  \item There is a formula a on the left and b on the right such that a and b are congruent.
+  \item There are two formulas a and !b on the left such that a and b are congruent.
+  \item There are two formulas a and !b on the right such that a and b are congruent.
+  \item The sequent is Ol-valid without equality reasoning
+\end{itemize}
+Note that congruence closure modulo OL is an open problem.
+
+\begin{example}
+  The following statements are provable by \lstinline|Congruence|:
+\newline\begin{lstlisting}[language=lisa, frame=single]
+val congruence1 = Theorem ((a === b, b === c) |- f(a) === f(c)) {
+  have(thesis) by Congruence
+}
+\end{lstlisting}
+
+\begin{lstlisting}[language=lisa, frame=single]
+val congruence2 = Theorem (
+  (F(F(F(F(F(F(F(x))))))) === x, F(F(F(F(F(x))))) === x) 
+  |- (F(x) === x)
+) {
+  have(thesis) by Congruence
+}
+\end{lstlisting}
+
+\begin{lstlisting}[language=lisa, frame=single]
+val congruence3 = Theorem (
+  (a === b, b === c, P(f(c)) <=> Q, P(f(a))) 
+  |- Q
+) {
+  have(thesis) by Congruence
+}
+\end{lstlisting}
+  
+\end{example}
+
+The tactic computes the congruence closure of all terms and formulas, with respect to the given equalities and equivalences, using an egraph datastructure \cite{willseyEggFastExtensible2021, nelsonFastDecisionProcedures1980}. The egraph contains two union-find datastructure which maintain equivalence classes of formulas and terms, respectively. The union-finds are equiped with an explain method, which can output a path of equalities between any two points in the same equivalence class, as in \cite{nelsonFastDecisionProcedures1980}. Each such equality can come from the left hand-side of the sequent being proven (we call those \textit{external equalities}), or be consequences of congruence. For an equality labelled by a congruence, the equalities between all children terms can recursively be explained.
+
+\begin{example}
+  Consider again the sequent
+  $$
+  a = b, b = c \vdash f(a) = f(c)
+  $$
+  the domain of our egraph is$\lbrace a, b, c, f(a), f(c) \rbrace$. When $a$ and $b$ are merged and then $b$ and $c$ are emrged, the egraph detects that $f(a)$ and $f(c)$ are congruent and should also be merged. The explanation of $f(a) = f(c)$ is then \lstinline|Congruence($f(a)$, $f(c)$)|, and the explanation of $a = c$ is \lstinline|External($a$, $b$), External($b$, $c$)|.
+\end{example}
+
+Once the congruence closure is computed, the tactic checks if the sequent is satisfies any of the above conditions and returns a proof if it does (and otherwise fails).
+
 \subsection*{Goeland}
 Goeland\cite{DBLP:conf/cade/CaillerRDRB22} is an Automated Theorem prover for first order logic. The Goeland tactic exports a statement in SC-TPTP format, and call Goeland to prove it. Goeland produce a proof file in the SC-TPTP format, from which Lisa rebuilds a kernel proof.
 \paragraph*{Usage}.
-- 
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