diff --git a/.gitignore b/.gitignore index b05d0969a..ad6f38cb3 100644 --- a/.gitignore +++ b/.gitignore @@ -22,3 +22,6 @@ cache # emacs backup files *~ +hol-import/project/project/metals.sbt +hol-import/project/metals.sbt +project/project/metals.sbt diff --git a/build.sbt b/build.sbt index 8178b5599..d8b7eb176 100644 --- a/build.sbt +++ b/build.sbt @@ -20,7 +20,7 @@ ThisBuild / scalafixDependencies += "com.github.liancheng" %% "organize-imports" val commonSettings = Seq( - crossScalaVersions := Seq("2.12.13", "2.13.4", "3.0.1", "3.2.0"), + crossScalaVersions := Seq("3.3.1"), run / fork := true ) @@ -40,10 +40,11 @@ val commonSettings3 = commonSettings ++ Seq( // "-source:future", re-enable when liancheng/scalafix-organize-imports#221 is fixed ), + javaOptions += "-Xmx10G", libraryDependencies += "org.scalatest" %% "scalatest" % "3.2.10" % "test", libraryDependencies += "com.lihaoyi" %% "sourcecode" % "0.3.0", //libraryDependencies += "org.scala-lang.modules" %% "scala-parser-combinators" % "2.1.1", - libraryDependencies += ("io.github.uuverifiers" %% "princess" % "2023-06-19").cross(CrossVersion.for3Use2_13), + // libraryDependencies += ("io.github.uuverifiers" %% "princess" % "2023-06-19").cross(CrossVersion.for3Use2_13), Test / parallelExecution := false ) @@ -59,6 +60,8 @@ lazy val silex = githubProject("https://github.com/epfl-lara/silex.git", "fc07a8 scallion/scalacOptions ~= (_.filterNot(Set("-Wvalue-discard"))) silex/scalacOptions ~= (_.filterNot(Set("-Wvalue-discard"))) +lazy val holimp = RootProject(file("./hol-import/")) + lazy val root = Project( id = "lisa", base = file(".") @@ -81,6 +84,7 @@ lazy val sets = Project( base = file("lisa-sets") ) .settings(commonSettings3) + .dependsOn(holimp) .dependsOn(kernel, withTests(utils)) lazy val utils = Project( diff --git a/hol-import/build.sbt b/hol-import/build.sbt new file mode 100644 index 000000000..57215bab3 --- /dev/null +++ b/hol-import/build.sbt @@ -0,0 +1,14 @@ +val scala3Version = "3.3.1" + +lazy val root = project + .in(file(".")) + .settings( + name := "holimp", + version := "0.1.0-SNAPSHOT", + + scalaVersion := scala3Version, + + libraryDependencies += "org.scalameta" %% "munit" % "0.7.29" % Test, + libraryDependencies += "com.lihaoyi" %% "upickle" % "3.2.0", + libraryDependencies += "org.scala-lang.modules" %% "scala-parser-combinators" % "2.3.0" + ) diff --git a/hol-import/project/build.properties b/hol-import/project/build.properties new file mode 100644 index 000000000..04267b14a --- /dev/null +++ b/hol-import/project/build.properties @@ -0,0 +1 @@ +sbt.version=1.9.9 diff --git a/hol-import/src/main/scala/holimp/Core.scala b/hol-import/src/main/scala/holimp/Core.scala new file mode 100644 index 000000000..b16ec4172 --- /dev/null +++ b/hol-import/src/main/scala/holimp/Core.scala @@ -0,0 +1,67 @@ +package holimp + +import java.awt.Window.Type + +object Core: + + case class ApplicationOnNonFunctionException(term: Term) extends Exception(s"Constructed a term applying to a non-function.\n\tTerm: $term") + + sealed trait Type + case class TypeVariable(name: String) extends Type + case class TypeApplication(name: String, args: List[Type]) extends Type + + val BoolType = TypeApplication("bool", Nil) + + object FunType: + def apply(in: Type, out: Type): Type = + TypeApplication("fun", List(in, out)) + def unapply(t: Type): Option[(Type, Type)] = + t match + case TypeApplication("fun", List(in, out)) => Some((in, out)) + case _ => None + + sealed trait Term: + def tpe: Type + + case class Variable(name: String, tpe: Type) extends Term + case class Constant(name: String, tpe: Type) extends Term + case class Combination(left: Term, right: Term) extends Term: + def tpe: Type = + left.tpe match + case FunType(_, out) => out + case _ => throw ApplicationOnNonFunctionException(this) + + case class Abstraction(absVar: Variable, inner: Term) extends Term: + def tpe: Type = + FunType(absVar.tpe, inner.tpe) + + sealed trait ProofStep + case class REFL(term: Term) extends ProofStep + case class TRANS(left: ProofStep, right: ProofStep) extends ProofStep + case class MK_COMB(left: ProofStep, right: ProofStep) extends ProofStep + case class ABS(absVar: Variable, from: ProofStep) extends ProofStep + case class BETA(term: Term) extends ProofStep + case class ASSUME(term: Term) extends ProofStep + case class EQ_MP(left: ProofStep, right: ProofStep) extends ProofStep + case class DEDUCT_ANTISYM_RULE(left: ProofStep, right: ProofStep) extends ProofStep + case class INST(from: ProofStep, insts: Map[Variable, Term]) extends ProofStep + case class INST_TYPE(from: ProofStep, insts: Map[TypeVariable, Type]) extends ProofStep + case class AXIOM(term: Term) extends ProofStep + case class DEFINITION(name: String, term: Term) extends ProofStep + case class TYPE_DEFINITION(name: String, term: Term, just: ProofStep) extends ProofStep + + extension (t : Type) def pretty: String = + t match + case BoolType => "B" + case FunType(from, to) => s"(${from.pretty} => ${to.pretty})" + case TypeVariable(name) => name + case TypeApplication(name, args) => s"$name[${args.mkString(", ")}]" + + extension (t: Term) def pretty: String = + t match + case Variable(name, tpe) => s"$name : ${tpe.pretty}" + case Constant(name, tpe) => s"$name : ${tpe.pretty}" + case Combination(left, right) => s"(${left.pretty}) (${right.pretty})" + case Abstraction(absVar, inner) => s"\\${absVar.pretty}. ${inner.pretty}" + +end Core diff --git a/hol-import/src/main/scala/holimp/Main.scala b/hol-import/src/main/scala/holimp/Main.scala new file mode 100644 index 000000000..d34a55436 --- /dev/null +++ b/hol-import/src/main/scala/holimp/Main.scala @@ -0,0 +1,94 @@ +package holimp + +import Core.* +import upickle.default.{ReadWriter => RW, *} +import Parser._ +import upickle.default +import java.io.File + +object JSONParser: + case class SplitProofStep(name: String, dep1: Int, dep2: Int, strdep: String, termdep: String, termsdeps: List[List[String]], typesdeps: List[List[String]]) + + case class RawStep(id: Int, pr: SplitProofStep) + + given rws: RW[SplitProofStep] = macroRW + given rw: RW[RawStep] = macroRW + + val fileBase = "/home/sguilloud/Lisa/lisa/proofs/prooftrace2" + + def readJSON(file: File) = + read[Array[RawStep]](file) + + case class UnknownProofStepException(name: String) extends Exception + case class CouldNotParseException(msg: String, next: String) extends Exception(s"Could not parse\n\tMessage: $msg\n\tNext: $next") + case object IncompleteParsingException extends Exception + case object UnreachableCaseException extends Exception + + + extension [T] (res: ParseResult[T]) + def getDone = + res match + case Success(out, next) if next.atEnd => out + case Success(_, _) => throw IncompleteParsingException + case NoSuccess(msg, next) => throw CouldNotParseException(msg, next.toString()) + case _ => throw UnreachableCaseException + + + def parseTerm(str: String): Term = parse(term, str).getDone + + def parseAsVar(str: String): Variable = parse(variable, str).getDone + + def parseInst(insts: List[List[String]]): Map[Variable, Term] = + insts.map{ case List(left, right) => parseAsVar(left) -> parseTerm(right); case _ => throw UnreachableCaseException}.toMap + + def parseTypeInst(insts: List[List[String]]): Map[TypeVariable, Type] = + def parseAsTVar(str: String) = parse(typeVariable, str).getDone + def parseType(str: String) = parse(typ, str).getDone + insts.map{ case List(left, right) => parseAsTVar(left) -> parseType(right); case _ => throw UnreachableCaseException}.toMap + + + val stepDict = Map.empty[Int, ProofStep] + + def processSplitStep(step: SplitProofStep, map: Map[Int, ProofStep]): ProofStep = + step.name match + case "REFL" => REFL(parseTerm(step.termdep)) + case "TRANS" => TRANS(map(step.dep1), map(step.dep2)) + case "MK_COMB" => MK_COMB(map(step.dep1), map(step.dep2)) + case "ABS" => ABS(parseAsVar(step.termdep), map(step.dep1)) + case "BETA" => BETA(parseTerm(step.termdep)) + case "ASSUME" => ASSUME(parseTerm(step.termdep)) + case "EQ_MP" => EQ_MP(map(step.dep1), map(step.dep2)) + case "DEDUCT_ANTISYM_RULE" => DEDUCT_ANTISYM_RULE(map(step.dep1), map(step.dep2)) + case "INST" => INST(map(step.dep1), parseInst(step.termsdeps)) + case "INST_TYPE" => INST_TYPE(map(step.dep1), parseTypeInst(step.typesdeps)) + case "AXIOM" => AXIOM(parseTerm(step.termdep)) + case "DEFINITION" => DEFINITION(step.strdep, parseTerm(step.termdep)) + case "TYPE_DEFINITION" => TYPE_DEFINITION(step.strdep, parseTerm(step.termdep), map(step.dep1)) + case _ => throw UnknownProofStepException(step.name) + + + def getProofs = + val rawSteps = readJSON(File(s"$fileBase.proofs")) + val mapped = rawSteps.foldLeft(Map.empty[Int, ProofStep]) { case (map, RawStep(i, step)) => + val readStep = processSplitStep(step, map) + map + (i -> readStep) + } + mapped + + case class TheoremRef(id: Int, nm: String) + + def getTheorems = + given rwt: RW[TheoremRef] = macroRW + read[List[TheoremRef]](File(s"$fileBase.names")) + + def getStatements = + case class Sequent(hy: List[String], cc: String) + case class TheoremStatement(id: Int, th: Sequent) + given rwns: RW[Sequent] = macroRW + given rwn: RW[TheoremStatement] = macroRW + read[Array[TheoremStatement]](File(s"$fileBase.theorems")).map(th => th.id -> (th.th.hy.map(parseTerm), parseTerm(th.th.cc))).toMap + +@main def hello(): Unit = + // val m = JSONParser.getProofs + val thm = JSONParser.getStatements(188) + println(thm._2.pretty) diff --git a/hol-import/src/main/scala/holimp/Parser.scala b/hol-import/src/main/scala/holimp/Parser.scala new file mode 100644 index 000000000..d08f514c9 --- /dev/null +++ b/hol-import/src/main/scala/holimp/Parser.scala @@ -0,0 +1,25 @@ +package holimp + +import Core.* +import scala.util.parsing.combinator.* + +object Parser extends RegexParsers: + type Identifier = String + + def isParen(c: Char): Boolean = c == '[' || c == ']' || c == '(' || c == ')' + + def char: Parser[Char] = acceptIf(!isParen(_))(el => s"Unexpected: $el") + def identifier: Parser[Identifier] = rep1(char) ^^ {_.mkString} + def typ: Parser[Type] = typeVariable ||| typeApplication + def term: Parser[Term] = variable ||| constant ||| combination ||| abstraction + + def variable: Parser[Variable] = "v(" ~ identifier ~ ")(" ~ typ ~ ")" ^^ { case _ ~ id ~ _ ~ tp ~ _ => Variable(id, tp) } + def constant: Parser[Constant] = "c(" ~ identifier ~ ")(" ~ typ ~ ")" ^^ { case _ ~ id ~ _ ~ tp ~ _ => Constant(id, tp) } + def combination: Parser[Combination] = "C(" ~ term ~ ")(" ~ term ~ ")" ^^ { case _ ~ t1 ~ _ ~ t2 ~ _ => Combination(t1, t2) } + def abstraction: Parser[Abstraction] = "A(" ~ variable ~ ")(" ~ term ~ ")" ^^ { case _ ~ v ~ _ ~ tm ~ _ => Abstraction(v, tm) } + + def typeVariable: Parser[TypeVariable] = "t[" ~ identifier ~ "]" ^^ { case _ ~ id ~ _ => TypeVariable(id)} + def typeApplication: Parser[TypeApplication] = "T[" ~ identifier ~ "][" ~ rep(typeArg) ~ "]" ^^ { case _ ~ id ~ _ ~ args ~ _ => TypeApplication(id, args) } + def typeArg: Parser[Type] = "[" ~ typ ~ "]" ^^ { case _ ~ tp ~ _ => tp} + +end Parser \ No newline at end of file diff --git a/hol-import/src/main/scala/import.sc b/hol-import/src/main/scala/import.sc new file mode 100644 index 000000000..bc3c1a6c9 --- /dev/null +++ b/hol-import/src/main/scala/import.sc @@ -0,0 +1,94 @@ +//> using dep "com.lihaoyi::upickle:3.2.0" + +object Core: + + case class ApplicationOnNonFunctionException(term: Term) extends Exception(s"Constructed a term applying to a non-function.\n\tTerm: $term") + + sealed trait Type + case object Unit extends Type + case object Bool extends Type + case object Ind extends Type + case class Func(inputType: Type, outputType: Type) extends Type + case class TypeVariable(name: String) extends Type + case class CustomType(name:String) extends Type + + sealed trait Term: + def tpe: Type + case class Variable(name: String, tpe: Type) extends Term + case class Abstraction(x: Variable, inner: Term) extends Term: + def tpe: Type = Func(x.tpe, inner.tpe) + case class Application(fun: Term, arg: Term) extends Term: + def tpe: Type = + fun.tpe match + case Func(inputType, outputType) => outputType + case _ => throw ApplicationOnNonFunctionException(this) + case class Constant(name: String, tpe: Type, paramTypes: List[TypeVariable]) extends Term +end Core + +import Core.* + +import upickle.default.* +import upickle.default.{ReadWriter => RW, macroRW} + +case class SplitProofStep(name: String, dep1: Int, dep2: Int, strdep: String, termdep: String, termsdeps: List[List[String]], typesdeps: List[List[String]]) + +case class RawStep(id: Int, pr: SplitProofStep) + +given rws: RW[SplitProofStep] = macroRW +given rw: RW[RawStep] = macroRW + +sealed trait ProofStep +case class REFL(term: Term) extends ProofStep +case class TRANS(left: ProofStep, right: ProofStep) extends ProofStep +case class MK_COMB(left: ProofStep, right: ProofStep) extends ProofStep +case class ABS(absVar: Variable, from: ProofStep) extends ProofStep +case class BETA(from: ProofStep) extends ProofStep +case class ASSUME(term: Term) extends ProofStep +case class EQ_MP(left: ProofStep, right: ProofStep) extends ProofStep +case class DEDUCT_ANTISYM_RULE(left: ProofStep, right: ProofStep) extends ProofStep +case class INST(from: ProofStep, insts: Map[Variable, Term]) extends ProofStep +case class INST_TYPE(from: ProofStep, insts: Map[TypeVariable, Type]) extends ProofStep +case class AXIOM(term: Term) extends ProofStep +case class DEFINITION(name: String, term: Term) extends ProofStep +case class TYPE_DEFINITION(name: String, term: Term, just: ProofStep) extends ProofStep + +case class UnknownProofStepException(name: String) extends Exception + +def parseTerm(str: String): Term = + def rec(str: String): Option[Term] = + val leading = str(0) + ??? + rec(str).get + +def parseAsVar(str: String): Variable = ??? + +def parseInst(insts: List[List[String]]): Map[Variable, Term] = ??? + +def parseTypeInst(insts: List[List[String]]): Map[TypeVariable, Type] = ??? + +val stepDict = Map.empty[Int, ProofStep] + +def processSplitStep(using map: Map[Int, ProofStep])(step: SplitProofStep): ProofStep = + step.name match + case "REFL" => REFL(parseTerm(step.termdep)) + case "TRANS" => TRANS(map(step.dep1), map(step.dep2)) + case "MK_COMB" => MK_COMB(map(step.dep1), map(step.dep2)) + case "ABS" => ABS(parseAsVar(step.termdep), map(step.dep1)) + case "BETA" => BETA(map(step.dep1)) + case "ASSUME" => ASSUME(parseTerm(step.termdep)) + case "EQ_MP" => EQ_MP(map(step.dep1), map(step.dep2)) + case "DEDUCT_ANTISYM_RULE" => DEDUCT_ANTISYM_RULE(map(step.dep1), map(step.dep2)) + case "INST" => INST(map(step.dep1), parseInst(step.termsdeps)) + case "INST_TYPE" => INST_TYPE(map(step.dep1), parseTypeInst(step.typesdeps)) + case "AXIOM" => AXIOM(parseTerm(step.termdep)) + case "DEFINITION" => DEFINITION(step.strdep, parseTerm(step.termdep)) + case "TYPE_DEFINITION" => TYPE_DEFINITION(step.strdep, parseTerm(step.termdep), map(step.dep1)) + case _ => throw UnknownProofStepException(step.name) + +val rawSteps = read[Array[RawStep]](java.io.File("new.proofs")) +val mapped = rawSteps.foldLeft(stepDict) { case (map, RawStep(i, step)) => + val readStep = processSplitStep(using map)(step) + map + (i -> readStep) +} +println(mapped) + diff --git a/lisa-examples/src/main/scala/Test.scala b/lisa-examples/src/main/scala/Test.scala index 026a3761a..da8b2b350 100644 --- a/lisa-examples/src/main/scala/Test.scala +++ b/lisa-examples/src/main/scala/Test.scala @@ -1,3 +1,4 @@ +import lisa.maths.settheory.types.TypeLib.|=> object Test extends lisa.Main { @@ -53,5 +54,37 @@ object Test extends lisa.Main { have(thesis) by Restate.from(ax of (c, d, P := ===)) showCurrentProof() } +/* + { + val A = variable + val B = variable + val app = function[2] + val f = variable + val g = variable + val x = variable + val y = variable + val thm5 = Theorem(() |- ()) { + val s1 = have((f ∈ A, x ∈ B) |- (app(f, x) === app(f, x))) by Sorry + val s2 = have((f ∈ A, g ∈ A) |- (f===g)) by Sorry + val s3 = have((x ∈ B, y ∈ B) |- (x===y)) by Sorry + have((x ∈ B, y ∈ B, f ∈ A, g ∈ A) |- (app(f, x) === app(g, y))) by Substitution.ApplyRules(s2, s3)(s1) + } + } +*/ + +/* + { + val f = function[1] + val x = variable + val y = variable + val P = formulaVariable + val Q = predicate[1] + val thm6 = Theorem(() |- ()) { + val s1 = have (P |- (f(x) === f(y)) ) by Sorry + assume(Q(f(x))) + thenHave( Q(f(y)) ) by Substitution.ApplyRules(s1) + } + } + */ } diff --git a/lisa-sets/src/main/scala/lisa/Main.scala b/lisa-sets/src/main/scala/lisa/Main.scala index 3d21fb8f4..fdc891e06 100644 --- a/lisa-sets/src/main/scala/lisa/Main.scala +++ b/lisa-sets/src/main/scala/lisa/Main.scala @@ -27,6 +27,9 @@ trait Main extends BasicMain { def definition: JUSTIFICATION = { getDefinition(symbol).get } + def shortDefinition: JUSTIFICATION = { + getShortDefinition(symbol).get + } } } diff --git a/lisa-sets/src/main/scala/lisa/automation/CommonTactics.scala b/lisa-sets/src/main/scala/lisa/automation/CommonTactics.scala index 9430195a1..c5b8e80a2 100644 --- a/lisa-sets/src/main/scala/lisa/automation/CommonTactics.scala +++ b/lisa-sets/src/main/scala/lisa/automation/CommonTactics.scala @@ -113,7 +113,7 @@ object CommonTactics { case F.AppliedPredicate(F.`equality`, Seq(`x`, (y: F.Variable))) if x != y && F.contains(uniquenessSeq.left, phi.substitute(x := y)) => y - case F.AppliedPredicate(F.`equality`, List(F.AppliedFunction(y: F.Variable, _), F.AppliedFunction(`x`, _))) if x != y && F.contains(uniquenessSeq.left, phi.substitute(x := y)) => + case F.AppliedPredicate(F.`equality`, List(F.AppliedFunctional(y: F.Variable, _), F.AppliedFunctional(`x`, _))) if x != y && F.contains(uniquenessSeq.left, phi.substitute(x := y)) => y } match { case Some(y) => ExistenceAndUniqueness.withParameters(phi, x, y)(existence, uniqueness)(bot) diff --git a/lisa-sets/src/main/scala/lisa/hol/HOL.scala b/lisa-sets/src/main/scala/lisa/hol/HOL.scala new file mode 100644 index 000000000..3a753f465 --- /dev/null +++ b/lisa-sets/src/main/scala/lisa/hol/HOL.scala @@ -0,0 +1,36 @@ +package lisa + +import lisa.SetTheoryLibrary +import lisa.prooflib.BasicMain + +/** + * The parent trait of all theory files containing mathematical development + */ +trait HOL extends BasicMain { + export lisa.maths.settheory.types.TypeSystem.{`*` => _, *} + export lisa.maths.settheory.types.TypeLib.{any, Zero, One, |=>, |-} + export TypeChecker.* + export lisa.hol.VarsAndFunctions.{main => _, given, *} + export SetTheoryLibrary.{*, given} + val STL = SetTheoryLibrary + val F: lisa.fol.FOL.type = lisa.fol.FOL + export F.{Term, variable, ===, given} + + + + private val A = variable + + + def assume(using proof: library.Proof)(t: Term): proof.ProofStep = + proof.addAssumption(eqOne(t)) + val seq = HOLSequent(Set(), t) + val r = have(seq) by lisa.prooflib.BasicStepTactic.Hypothesis + r + + + def withCTX(s:F.Sequent): F.Sequent = + val ctx = computeContextOfFormulas(s.left ++ s.right) + s ++<< ((ctx._1 ++ ctx._2) |- ()) + + +} \ No newline at end of file diff --git a/lisa-sets/src/main/scala/lisa/hol/HOLImport.scala b/lisa-sets/src/main/scala/lisa/hol/HOLImport.scala new file mode 100644 index 000000000..c417152ba --- /dev/null +++ b/lisa-sets/src/main/scala/lisa/hol/HOLImport.scala @@ -0,0 +1,277 @@ +package lisa.hol + +import holimp.{Core => HOLL} +import lisa.utils.memoization.memoized + +import lisa.maths.settheory.types.TypeLib.definition +import HOLSteps.{_, given} +import lisa.prooflib.BasicStepTactic._ +import lisa.prooflib.SimpleDeducedSteps._ +import lisa.automation.Substitution +import lisa.fol.FOL as F +import lisa.fol.FOL.{Identifier, Term} +import lisa.fol.FOL.{*, given} +import lisa.automation._ +import sourcecode.Name +import lisa.utils.Serialization.termLabelToString +import lisa.utils.unification.UnificationUtils.matchTerm +import lisa.maths.settheory.types.TypeSystem +import lisa.hol.VarsAndFunctions.TypingTheorem + +object HOLImport extends lisa.HOL { + + import library._ + + val parser = holimp.JSONParser + val steps = parser.getProofs + val thms = parser.getTheorems + val stmts = parser.getStatements + + private def toLisaType__(typ: HOLL.Type): Type = + typ match + // special cases + case HOLL.BoolType => 𝔹 + case HOLL.FunType(from, to) => toLisaType(from) |=> toLisaType(to) + // others + case HOLL.TypeVariable(name) => HOLSteps.variable(using name) + case HOLL.TypeApplication(name, args) => ??? + + case object ExpectedVariableException extends Exception + + private def asVar(v: Term): TypedVar = + v match + case v : TypedVar => v + case _ => throw ExpectedVariableException + + + private def toLisaTerm__(term: HOLL.Term): Term = + term match + case HOLL.Variable(name, typ) => typedvar(using name)(toLisaType(typ)) + // special case equality + case HOLL.Combination( + HOLL.Combination(HOLL.Constant("=", _), l), + r + ) => toLisaTerm(l) =:= toLisaTerm(r) + case HOLL.Constant(name, typ) => Constants.get(name, toLisaType(typ)) + case HOLL.Combination(left, right) => toLisaTerm(left)*(toLisaTerm(right)) + case HOLL.Abstraction(vbl, tm) => λ(asVar(toLisaTerm(vbl)), toLisaTerm(tm)) + + val toLisaType: HOLL.Type => Type = memoized(toLisaType__) + val toLisaTerm: HOLL.Term => Term = memoized(toLisaTerm__) + + object Constants: + sealed trait LabelStore + case class JustConstant[A <: TypeSystem.Class](c: TypedConstant[A]) extends LabelStore + case class Functional[N <: Arity](f: TypedConstantFunctional[N], freeType: F.Term, params: Seq[F.Variable], innerDef: JUSTIFICATION) extends LabelStore + + case object MalformedTypeInstantiationException extends Exception + + private val illegalChars = "}]`)[{(,;?_." + private val subst = illegalChars.zipWithIndex.toMap.view.mapValues(c => (9312 + c).toChar) + + def sanitizeName(name: String): String = + name.map(c => if subst.contains(c) then subst(c) else c) + + private val constants = scala.collection.mutable.HashMap.empty[String, LabelStore] + def register[A <: TypeSystem.Class](c: TypedConstant[A]): Unit = + // two things should not have the same name, as they cannot be distinguished by no. from HOL + constants.update(c.id.name, JustConstant(c)) + def register[N <: Arity](c: TypedConstantFunctional[N], tpe: F.Term, params: Seq[F.Variable], innerDef: JUSTIFICATION): Unit = + // two things should not have the same name, as they cannot be distinguished by no. from HOL + constants.update(c.id.name, Functional(c, tpe, params, innerDef)) + def get(name: String, tpe: Term) = + // guaranteed to be safe if we read theorems in the order HOL produces them + val store = constants("HOL@" + sanitizeName(name)) + store match + case JustConstant(c) => c + case Functional(f, freeType, params, _) => + val subst = matchTerm(tpe, freeType) + if subst.isEmpty then + throw MalformedTypeInstantiationException + else + val substs = subst.get + val inputs = params.map(p => substs.getOrElse(p, p)) + f.applySeq(inputs) + + def getDefinition(name: String): JUSTIFICATION = + constants("HOL@" + sanitizeName(name)) match + case JustConstant(c) => c.definition + case Functional(_, _, _, innerDef) => + // the definition, instantiated into a usable form at time of construction + innerDef + + + // handling equality separately + val equality = Functional(=:=, (A |=> (A |=> B)), Seq(A), eqCorrect) + constants.update("HOL@=", equality) + + + import Constants.{register, get, getDefinition, sanitizeName} + + private val theoremCache = collection.mutable.HashMap.empty[HOLL.ProofStep, library.THM] + + private def reconstruct(using library: HOLSteps.library.type, ctx: library.Proof)(proof: HOLL.ProofStep): ctx.Fact = + import HOLSteps._ + object Rec: + val rec = (p: HOLL.ProofStep) => theoremCache.getOrElse(p, recm(p)) + val recm: HOLL.ProofStep => ctx.ProofStep = memoized(rec_(_)) + def rec_(proof: HOLL.ProofStep): ctx.ProofStep = { + def transformed: ctx.ProofTacticJudgement = + proof match + case HOLL.REFL(term) => REFL(toLisaTerm(term)) + case HOLL.TRANS(left, right) => + val l = rec(left) + val r = rec(right) + TRANS(l, r) + case HOLL.MK_COMB(left, right) => + val l = rec(left) + val r = rec(right) + MK_COMB(l, r) + case HOLL.ABS(absVar, from) => + val pf = rec(from) + ABS(asVar(toLisaTerm(absVar)))(pf) + case HOLL.BETA(term) => + val inp = toLisaTerm(term) + val fin = BETA(inp) + fin + case HOLL.ASSUME(term) => ASSUME(toLisaTerm(term)) + case HOLL.EQ_MP(left, right) => + val pf = rec(left) + EQ_MP(rec(left), rec(right)) + case HOLL.DEDUCT_ANTISYM_RULE(left, right) => DEDUCT_ANTISYM_RULE(rec(left), rec(right)) + case HOLL.INST(from, insts) => + val inner = rec(from) + val instss = insts.toSeq.map((k, v) => asVar(toLisaTerm(k)) -> toLisaTerm(v)) + val fin = INST(instss, inner) + fin + case HOLL.INST_TYPE(from, insts) => + def singleTypeInst = (step: ctx.Fact, inst: (HOLL.TypeVariable, HOLL.Type)) => + val x = + toLisaType(inst._1) match + case v : F.Variable => v + case _ => throw ExpectedVariableException + val typ = toLisaType(inst._2) + library.have(INST_TYPE(x, typ, step)) + val fin = insts.foldLeft(rec(from))(singleTypeInst) + Restate.from(fin)(fin.statement) + case HOLL.AXIOM(term) => + // prove the axioms and simply retrieve them + ??? + case HOLL.DEFINITION(name, term) => + // must have been handled previously + // retrieve it + val defn = Constants.getDefinition(name) + Restate.from(defn)(defn.statement) + case HOLL.TYPE_DEFINITION(name, term, just) => + // is this ever used? + // what does it look like? + ??? + + val tr = library.have(transformed) + tr + } + + val res = Rec.rec(proof) + res + + /** + * Checks if this HOL Light sequent is a "new_basic_definition". + * + * Must be of the form DEFINITION(`|- ((=) (symbol)) (defn)`) if we have not + * seen `symbol` before. + * + * The form and visibility invariants are assumed to be maintained by the HOL + * system for now. + * + */ + private def isDefinition(proof: HOLL.ProofStep): Boolean = proof.isInstanceOf[HOLL.DEFINITION] + + case class MalformedDefinitionException(id: Int, term: HOLL.Term) extends Exception(s"Malformed definition at id $id: ${term.pretty}") + case class MalformedDefinitionFormat(id: Identifier) extends Exception(s"Definition of $id is not of the form forall(v, (v = $id) <=> (context => v = term))") + + val lisaThms = + for + thm <- thms.sortBy(_.id) + yield + println(s"Processing ${thm.id}") + val (hypotheses, conclusion) = stmts(thm.id) + val proof = steps(thm.id) + if isDefinition(proof) then + // register the constant + assert(hypotheses.isEmpty) + import HOLL.{Combination, Constant} + conclusion match + case Combination(Combination(Constant("=", _), Constant(name, typ)), defTerm) => + val term = toLisaTerm(defTerm) + val tpe = toLisaType(typ) + val freeTypes = tpe.freeVariables.toSeq + // TODO: special case freeTypes.isEmpty + val context = computeContext(Set(term)) + // we need to check the set of declared abstractions in this term, and totally order and quantify over them + val orderedAbstractions: List[Variable] = + val abstractions = context._2 + // for each abstraction, first find which variable it's defining + // then, find everything it depends on + val dependencies = abstractions.map( + abs => + val l = abs.args(0).asInstanceOf[TypeAssignment[Type]].t + assert(l.isInstanceOf[Variable]) + l.asInstanceOf[Variable & Term] -> (abs.bodyProp.allSchematicLabels.filter(a => a.id.name.startsWith("$λ") && a != l.label && a.isInstanceOf[Variable]).toList) + ).toMap + val ls = dependencies.keys.toList + ls.sortWith((l, r) => dependencies.getOrElse(l, Nil).contains(r)) + ls + val z = variable + val ctx = (context._1 ++ context._2).toSeq + inline def base(z: Term) = F.and(ctx) ==> (z === term) + inline def zDef(z: Term) = + orderedAbstractions.foldRight(base(z))((label, inner) => forall(label, inner)) + val just = Lemma(existsOne(z, zDef(z))) { + sorry + } + val newLabel = + FunctionDefinition(sanitizeName(s"HOL@$name"), thm.id, "HOLLight")(freeTypes, z, zDef(z), just).label + val baseTypingFormula: F.Formula = (newLabel.applySeq(freeTypes) :: tpe) + val quantifiedTypingFormula = freeTypes.foldRight(baseTypingFormula)((v, step) => forall(v, step)) + val typingProof = Lemma(quantifiedTypingFormula) { + // have(zDef(constant)) by InstantiateForall(constant)(constant.definition) + // val instDef = thenHave(base(constant)) by InstantiateForall(orderedAbstractions: _*) + // val typed = have(TypingTheorem(term :: tpe)) + // have(constant === term) by Tautology.from(instDef, typed) + // have(constant :: tpe) by Substitution.ApplyRules(lastStep)(typed) + sorry + } + val typedLabel = TypedConstantFunctional(newLabel.id, newLabel.arity, FunctionalClass(freeTypes.map(x => any), freeTypes, tpe, newLabel.arity), typingProof) + val f = typedLabel.applySeq(freeTypes) + val innerDef = Lemma((f =:= term)) { + val typingProof = have(ProofType(term)) + val fTyping = have(ProofType(f)) + + if !ctx.isEmpty then assume(ctx: _*) + have(zDef(f)) by Weakening(newLabel.definition of f) + have(base(f)) by Weakening(lastStep.of(orderedAbstractions: _*)) + thenHave(f === term) by Weakening + have(thesis) by Tautology.from(lastStep, fTyping, typingProof, eqCorrect of (A -> tpe, x -> f, y -> term)) + } + println(s"DEFINING ${innerDef.statement}") + Constants.register(typedLabel, tpe, freeTypes, innerDef) + newLabel.definition + case _ => throw MalformedDefinitionException(thm.id, conclusion) + else + val lisaHyps = hypotheses.toSet.map(toLisaTerm) + val lisaConc = toLisaTerm(conclusion) + val sequent = HOLSequent(lisaHyps, lisaConc) + val proof = steps(thm.id) + + val res = THM(sequent, thm.nm, thm.id, "HOL.Import", Theorem): + val step = reconstruct(proof) + have(Clean.all(step)) + println(s"PROVED :: \n\t${lastStep.statement}\nNEEDED ::\n\t$sequent") + + theoremCache.update(proof, res) + res + + @main + def importHOLLight = + lisaThms.foreach(t => println(t.repr)) +} diff --git a/lisa-sets/src/main/scala/lisa/hol/HOLSteps.scala b/lisa-sets/src/main/scala/lisa/hol/HOLSteps.scala new file mode 100644 index 000000000..1d632f560 --- /dev/null +++ b/lisa-sets/src/main/scala/lisa/hol/HOLSteps.scala @@ -0,0 +1,1040 @@ +package lisa.hol +import lisa.fol.FOL as F +import lisa.prooflib.BasicStepTactic.* +import lisa.prooflib.SimpleDeducedSteps.* +import lisa.prooflib.ProofTacticLib.* +import lisa.automation.* + +import lisa.maths.settheory.SetTheory.{singleton, app} +import lisa.kernel.proof.SCProofChecker.checkSCProof +import lisa.utils.KernelHelpers.checkProof +import lisa.fol.FOLHelpers.freshVariable +import lisa.utils.Serialization.instSchema +import lisa.prooflib.BasicStepTactic +import lisa.prooflib.SimpleDeducedSteps.Discharge +import lisa.automation.Tableau.pr +import lisa.maths.settheory.SetTheory.definition +import lisa.maths.settheory.SetTheory.singletonNonEmpty + +/** + * Here we define and implement all the basic steps from HOL Light + */ +object HOLSteps extends lisa.HOL { + import lisa.SetTheoryLibrary.* + + //draft() + + //REFL + //_TRANS + //MK_COMB + //ABS + //BETA + //ETA + //ASSUME + //_EQ_MP + //DEDUCT_ANTISYM_RULE + //INST + //INST_TYPE + + val A = variable + val B = variable + val v = typedvar(B) + val w = typedvar(A) + val x = typedvar(A) + val y = typedvar(A) + val z = typedvar(A) + val e = typedvar(A |=> A) + val f = typedvar(A |=> B) + val g = typedvar(A |=> B) + val h = typedvar(B |=> A) + + val p = typedvar(𝔹) + val q = typedvar(𝔹) + val r = typedvar(𝔹) + + val eqCorrect = Theorem((x::A, y::A) |- ((x =:= y)===One) <=> (x===y)) { + have(thesis) by Restate.from(eqDefin) + } + val eqRefl = Theorem((x =:= x)) { + have(x::A |- (x === x)) by Restate + thenHave(x =:= x) by Substitution.ApplyRules(eqCorrect of (HOLSteps.y := x)) + + } + + val eqTrans = Theorem( ((x =:= y), (y =:= z)) |- (x =:= z) ) { + have((x::A, y::A, z::A, x === y, y === z)|- (x === y)) by Restate + thenHave((x::A, y::A, z::A, x === y, y === z)|- (x === z)) by Substitution.ApplyRules(y === z) + thenHave(((x::A, y::A, z::A, eqOne(x =:= y), y === z)|- (x === z))) by Substitution.ApplyRules(eqCorrect) + thenHave(((x::A, y::A, z::A, eqOne(x =:= y), eqOne(y =:= z))|- (x === z))) by Substitution.ApplyRules(eqCorrect of (x := y, y := z)) + thenHave(((x::A, y::A, z::A, eqOne(x =:= y), eqOne(y =:= z))|- eqOne(x =:= z))) by Substitution.ApplyRules(eqCorrect of (y := z)) + } + + + val eqSym = Theorem( (x =:= y) |- (y =:= x) ) { + sorry + } + + val funcUnique = Theorem((f :: (A |=> B), g :: (A |=> B), tforall(x, f*x === g*x)) |- (f === g)) {have(thesis) by Restate.from(functionalExtentionality)} + val funcUnique2 = Lemma((f :: (A |=> B), g :: (A |=> B), tforall(x, f*x === g*x)) |- ((f =:= g) === One)) { + have(thesis) by Substitution.ApplyRules(eqCorrect of (HOLSteps.x := f, HOLSteps.y := g, A := (A |=> B)))(funcUnique) + } + + val Bdef = Theorem((x ∈ 𝔹) |- ((x === Zero) \/ (x === One))) { + val s1 = have ((x ∈ unorderedPair(∅, singleton(∅))) |- ((x === ∅ )\/ (x === singleton(∅)))) by Weakening(pairAxiom of (z := x, x := ∅ , y := singleton(∅))) + val beq = have(unorderedPair(∅, singleton(∅)) === 𝔹) by Weakening(𝔹.definition of 𝔹) + val s2 = have((x ∈ 𝔹) |- ((x === ∅) \/ (x === singleton(∅)))) by Substitution.ApplyRules(beq)(s1) + val zeq = have(∅ === Zero) by Weakening(Zero.definition of Zero) + val s3 = have((x ∈ 𝔹) |- ((x === Zero) \/ (x === singleton(∅)))) by Substitution.ApplyRules(zeq)(s2) + val oeq = have(singleton(∅) === One) by Weakening(One.definition of One) + have((x ∈ 𝔹) |- ((x === Zero) \/ (x === One))) by Substitution.ApplyRules(oeq)(s3) + } + + val propExt = Theorem((p :: 𝔹, q :: 𝔹, (q === One) <=> (p === One)) |- (p === q)) { + + val h2 = have((p :: 𝔹, q :: 𝔹, (q === One) <=> (p === One), p === One) |- (p === One)) by Restate + val h3 = have((p :: 𝔹, q :: 𝔹, (q === One) <=> (p === One), p === One) |- (q === One)) by Restate + val h4 = have((p :: 𝔹, q :: 𝔹, (q === One) <=> (p === One), p === One) |- (p === q)) by Substitution.ApplyRules(h3)(h2) + + val neq = have ((p === Zero, p === One )|- ()) subproof { + val zeq = have(∅ === Zero) by Weakening(Zero.definition of Zero) + val oeq = have(singleton(∅) === One) by Weakening(One.definition of One) + have(∅ ∈ singleton(∅)) by Weakening(pairAxiom of (x:= ∅, y := ∅, z := ∅)) + have((∅ === singleton(∅)) |- ()) by Restate.from(singletonNonEmpty of (x:= ∅)) + + thenHave ((p === singleton(∅), p === ∅) |- ()) by Substitution.ApplyRules(p === ∅) + thenHave ((p === singleton(∅), p === Zero) |- ()) by Substitution.ApplyRules(zeq) + thenHave ((p === One, p === Zero) |- ()) by Substitution.ApplyRules(oeq) + } + val i1 = have((p :: 𝔹 |- (!(p === One)) <=> (p === Zero))) by Tautology.from(Bdef of ( x:= p), neq) + val i2 = have((p :: 𝔹, q :: 𝔹, (q === One) <=> (p === One)) |- !(q === One) <=> !(p === One)) by Tautology + val i3 = have((p :: 𝔹, q :: 𝔹, (q === One) <=> (p === One)) |- (q === Zero) <=> (p === Zero)) by Tautology.from(i2, i1, i1 of (p:=q)) + + val j2 = have ((p :: 𝔹, q :: 𝔹, (q === One) <=> (p === One), !(p === One), (q === Zero) <=> (p === Zero)) |- p === Zero) by Tautology.from(Bdef of (x := p)) + val j3 = have((p :: 𝔹, q :: 𝔹, (q === One) <=> (p === One), !(p === One), (q === Zero) <=> (p === Zero)) |- q === Zero) by Tautology.from(lastStep) + val j4 = have((p :: 𝔹, q :: 𝔹, (q === One) <=> (p === One), !(p === One), (q === Zero) <=> (p === Zero)) |- (p === q)) by Substitution.ApplyRules(j3)(j2) + + have(thesis) by Tautology.from(j4, i3, h4) + + } + + + + /** + * ------------------ + * |- t = t + */ + object REFL extends ProofTactic { + def apply(using proof: Proof)(t: Term): proof.ProofTacticJudgement = TacticSubproof{ + val s1 = have(ProofType(t)) //t::A + val typ = s1.statement.right.head.asInstanceOf[TypeAssignment[Term]].typ + have(holeq(typ)*t*t) by Tautology.from(eqRefl of (x := t, A := typ), s1) + + } + } + + /** + * |- s = t |- t = u + * --------------------- + * |- s = u + */ + object _TRANS extends ProofTactic { + def apply(using proof: Proof)(t1: proof.Fact, t2: proof.Fact): proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + val s1 = t1.statement + val s2 = t2.statement + (s1, s2) match { + case (HOLSequent(left1, =:=(aa)*s*ta), HOLSequent(left2, =:=(ab)*tb*u) ) => //equality is too strict + if ta == tb then + if aa == ab then + (s1.left ++ s2.left).foreach(assume(_)) + val p0 = have(s1) by Weakening(t1) + val r0 = have(((s :: aa), (ta :: aa), (u :: aa), (ta =:= u) === One) |- (s =:= u) === One) by Cut(p0, eqTrans of (x := s, y := ta, z := u, A := aa)) + val r1 = have(((s :: aa), (ta :: aa), (u :: aa)) |- (s =:= u) === One) by Cut(t2, r0) + val r2 = have(Discharge(have(ProofType(s)))(r1)) + val r3 = have(Discharge(have(ProofType(ta)))(r2)) + val r4 = have(Discharge(have(ProofType(u)))(r3)) + ip.cleanAssumptions + have(Clean.all(r4)) + else + return proof.InvalidProofTactic(s"Types don't agree: $aa and $ab") + else + return proof.InvalidProofTactic(s"Middle elements don't agree: $ta and $tb") + + case (HOLSequent(left1, right1), HOLSequent(left2, right2) ) => + return proof.InvalidProofTactic(s"The facts should have equalities") + case _ => + s1 match + case HOLSequent(left1, right1) => + return proof.InvalidProofTactic(s"The second fact is not parsable as an HOL sequent") + case _ => + return proof.InvalidProofTactic(s"The first fact is not parsable as an HOL sequent") + + } + + } + } + + /** + * Apply transitivity of equality, but with alpha equivalence. + */ + object TRANS extends ProofTactic { + def apply(using proof: Proof)(t1: proof.Fact, t2: proof.Fact): proof.ProofTacticJudgement = + val s1 = t1.statement + val s2 = t2.statement + + (s1, s2) match { + case (HOLSequent(left1, =:=(aa)*s*ta), HOLSequent(left2, =:=(ab)*tb*u) ) => //equality is too strict + if aa == ab then + if ta == tb then + _TRANS(t1, t2) + else + // try to see if ta alpha_eq tb + TacticSubproof: + val alpha = have(ALPHA_EQUIVALENCE(ta, tb)) + val s1 = have(_TRANS(t1, alpha)) + val s2 = have(_TRANS(s1, t2)) + else + return proof.InvalidProofTactic(s"Types don't agree: $aa and $ab") + + case (HOLSequent(left1, right1), HOLSequent(left2, right2) ) => + return proof.InvalidProofTactic(s"The facts should have equalities") + case _ => + s1 match + case HOLSequent(left1, right1) => + return proof.InvalidProofTactic(s"The second fact is not parsable as an HOL sequent") + case _ => + return proof.InvalidProofTactic(s"The first fact is not parsable as an HOL sequent") + + } + } + + /** + * |- f = g |- x = y + * --------------------- + * |- f x = g y + */ + object MK_COMB extends ProofTactic { + def apply(using proof: Proof)(f1: proof.Fact, f2: proof.Fact): proof.ProofTacticJudgement = TacticSubproof { + val fg = f1.statement + val xy = f2.statement + (fg, xy) match { + case (HOLSequent(left1, =:=(typ1)*f*g), HOLSequent(left2, =:=(typ2)*x*y) ) => //equality is too strict + typ1 match { + case |=>(`typ2`, b) => + (f1.statement.left ++ f2.statement.left).foreach(assume(_)) + val vx = typedvar(computeType(x)) + val vf = typedvar(computeType(f)) + val s1 = have(REFL(f*x)) + val s2 = have((f :: typ1, g::typ1) |- (f===g)) by Tautology.from(f1, eqCorrect of (HOLSteps.x := f, HOLSteps.y := g, A := typ1)) + val s3 = have((x :: typ2, y::typ2) |- (x===y)) by Tautology.from(f2, eqCorrect of (HOLSteps.x := x, HOLSteps.y := y, A := typ2)) + val s4 = have((x === y) |- (f*x =:= f*y) === One) by RightSubstEq.withParametersSimple(List((x, y)), F.lambda(vx, f*x =:= f*vx))(s1) + val s5 = have(((x === y), (f === g)) |- (f*x =:= g*y) === One) by RightSubstEq.withParametersSimple(List((f, g)), F.lambda(vf, f*x =:= vf*y))(s4) + val s6 = have((x :: typ2, y::typ2, (f === g)) |- (f*x =:= g*y) === One) by Cut(s3, s5) + val s7 = have((x :: typ2, y::typ2, f :: typ1, g::typ1) |- (f*x =:= g*y) === One) by Cut(s2, s6) + val d1 = have( Discharge(have(ProofType(x)))(s7) ) + val d2 = have( Discharge(have(ProofType(y)))(d1) ) + val d3 = have( Discharge(have(ProofType(f)))(d2) ) + val d4 = have( Discharge(have(ProofType(g)))(d3) ) + + case _ => + return proof.InvalidProofTactic(s"Types don't agree: fun types are $typ1 and arg types are $typ2") + } + case _ => + return proof.InvalidProofTactic(s"The facts should be of the form f =:= g and x =:= y") + } + } + } + + /** + * |- t =:= u + * --------------------- + * |- λx. t =:= λx. u + * + */ + object ABS extends ProofTactic { + def apply(using proof: Proof)(x: TypedVar)(prem: proof.Fact): proof.ProofTacticJudgement = TacticSubproof { ip ?=> + val s1 = prem.statement + s1 match { + case HOLSequent(left, =:=(typ1)*t*u) => + (s1.left - (x :: x.typ)).foreach(assume(_)) + val lt = λ(x, t) + val lu = λ(x, u) + val lctx12 = computeContext(Set(lt, lu)) + val lctx = lctx12._1 ++ lctx12._2 + lctx.foreach(a => + assume(a)) + val typjudgt = have(ProofType(t)) + + val typjudgltx = have(ProofType(lt*x)) + + val typjudgu = have(ProofType(u)) + + val typjudglux = have(ProofType(lu*x)) + + + val ctx12 = computeContext(Set(t, u)) + val ctx = ctx12._1 ++ ctx12._2 + (ctx - (x:: x.typ)).foreach(a => assume(a)) + val ltx_lux = have((x::x.typ) |- (lt*x === lu*x)) subproof { iip ?=> + Seq(typjudgt, typjudgltx, typjudgu, typjudglux).foreach( a => + assume(a.statement.right.head) + ) + assume(x :: x.typ) + + val ltprop = assume(lt.defin.bodyProp) + val luprop = assume(lu.defin.bodyProp) + val bt1 = have(BETA(lt*x)) + val bt = have((x :: x.typ) |- ((lt*x =:= t) === One)) by Weakening(bt1) + val bth = { + val phi = F.formulaVariable + val sl = (lt*x =:= t) === One + val sr = lt*x === t + val bth0 = have((x :: x.typ, sl <=> sr ) |- (lt*x === t)) by RightSubstIff.withParametersSimple(List((sl, sr)), F.lambda(phi, phi))(bt) + have(Discharge(eqCorrect of (HOLSteps.x := lt*x, HOLSteps.y := t, A := typ1))(bth0)) + } + + val btc = lastStep.statement // x::x.typ |- lt(a)...(z)(x) = t + + val bu1 = have(BETA(lu*x)) + val bu = have((x :: x.typ) |- ((lu*x =:= u) === One)) by Weakening(bu1) + //val buh = have((x :: x.typ) |- (lu*x === u)) by Substitution.ApplyRules(eqCorrect of (HOLSteps.x := lu*x, HOLSteps.y := u, A := typ1))(bu) + val buh = { + val phi = F.formulaVariable + val sl = (lu*x =:= u) === One + val sr = lu*x === u + val buh0 = have((x :: x.typ, sl <=> sr) |- (lu*x === u)) by RightSubstIff.withParametersSimple(List(((lu*x =:= u) === One, lu*x === u)), F.lambda(phi, phi))(bu) + have(Discharge(eqCorrect of (HOLSteps.x := lu*x, HOLSteps.y := u, A := typ1))(buh0)) + } + val buc = lastStep.statement // x::x.typ |- lt(a)...(z)(x) = u + + val s0 = have(t===u) by Tautology.from(prem, eqCorrect of (HOLSteps.x := t, HOLSteps.y := u, A := typ1)) + val s1 = { + val xx = freshVariable(Seq(lt*x), "xx") + val s10 = have((x :: x.typ, t === u) |- (lt*x === u)) by RightSubstEq.withParametersSimple(List((t, u)), F.lambda(xx, lt*x === xx))(bth) + have(Discharge(s0)(s10)) + } + //val s1 = have((x :: x.typ) |- ((lt*x)===u)) by Substitution.ApplyRules(s0)(bth) + val s2 = { + val xx = freshVariable(Seq(lt*x), "xx") + val s20 = have((x :: x.typ, u === lu*x) |- (lt*x === lu*x)) by RightSubstEq.withParametersSimple(List((u, lu*x)), F.lambda(xx, lt*x === xx))(s1) + have(Discharge(buh)(s20)) + } + //val s2 = have((x :: x.typ) |- ((lt*x)===(lu*x))) by Substitution.ApplyRules(buh)(s1) + + + //by RightSubstEq.withParametersSimple(List((r, r2)), F.lambda(xx, t*x === xx))(i0) //Substitution.ApplyRules(def_red_r)(i0) + // val i2 = have(Discharge(def_red_r)(i1)) + + + Seq(typjudgt, typjudgltx, typjudgu, typjudglux).foreach( a => + have(Discharge(a)(lastStep)) + ) + } + val s2c = lastStep.statement.right.head // lt*x = lu*x + val is2 = have(((x::x.typ) ==> s2c)) by Restate.from(lastStep) + + val qs2 = have(tforall(x, (lt*x)===(lu*x))) by RightForall(is2) + val h1 = have((lt:: (x.typ |=> typ1), lu:: (x.typ |=> typ1)) |- ((lt =:= lu) === One)) by Tautology.from(funcUnique2 of (f := lt, g := lu, HOLSteps.x := x, A := x.typ, B := typ1), qs2) + + val h2 = have( Discharge(have(ProofType(lt)))(h1) ) + val h3 = have( Discharge(have(ProofType(lu)))(h2) ) + have(Clean.all(h3)) + + + case _ => + return proof.InvalidProofTactic(s"The fact should be of the form t =:= u") + } + } + } + + + + /** + * BETA_CONV((λx. t) u) produces |- (λx. t) x =:= t[x := u] + */ + object BETA_CONV extends ProofTactic { + def apply(using proof: Proof)(t: Term): proof.ProofTacticJudgement = TacticSubproof{ + t match + case (l:Abstraction)*(r: Term) => + val tp = computeType(r) + if tp == l.bound.typ then + val rhs = l.BETA.statement.right.head.asInstanceOf[F.AppliedPredicate].args(0).asInstanceOf[AppliedFunction].arg + val b = l.BETA of (l.bound := r) + val h = have(b.statement) by Weakening(b) // add any floating assumptions in + val h1 = have(Discharge(have(ProofType(r)))(h)) + h.statement.right.head match + case eqOne(r) => + //println(s"ASKING DEF RED FOR $r") + val def_red_r = have(DEF_RED(r)) // r === r2 + def_red_r.statement.right.head match + case `r` === r2 => + val x = typedvar(𝔹) + val s0 = have((h1.statement.left ++ def_red_r.statement.left) |- eqOne(r)) by Weakening(h1) + val s1 = have((h1.statement.left ++ def_red_r.statement.left + (r === r2)) |- eqOne(r2)) by RightSubstEq.withParametersSimple(List(r -> r2), F.lambda(x, eqOne(x)))(s0) + val s2 = have((h1.statement.left ++ def_red_r.statement.left) |- eqOne(r2)) by Cut(def_red_r, s1) + have(Clean.all(s2)) + case _ => + return proof.InvalidProofTactic(s"Beta definition has malformed shape. Expected term === One.") + case _ => + return proof.InvalidProofTactic(s"Beta definition has malformed shape. Expected term === One.") + else + return proof.InvalidProofTactic(s"Beta redex has malformed argument type. Expected type ${l.bound.typ}, got $tp .") + case _ => + return proof.InvalidProofTactic(s"The term should be of the form (λx. t) v") + + } + } + + /** + * BETA((λx. t) x) produces |- (λx. t) x =:= t + */ + object BETA extends ProofTactic { + def apply(using proof: Proof)(t: Term): proof.ProofTacticJudgement = TacticSubproof{ + t match + case (l:Abstraction)*(r: TypedVar) => + //println(s"BETA GOT AS INPUT $t") + // assure the right shape is present, and pass to the general case + have(BETA_CONV(t)) + case _ => + //println(t) + //println(Abstraction.cache) + return proof.InvalidProofTactic(s"The term should be of the form (λx. t) y") + + } + } + + object BETA_PRIM extends ProofTactic { + def apply(using proof: Proof)(t: Term): proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + t match + case (l:Abstraction)*(r: TypedVar) if l.bound == r => + val b = l.BETA + val s1 = have(b.statement) by Weakening(b) //l*r =:= l.body + val ctx = computeContext(Set(l*r, l.body)) + ctx._1.foreach(a => assume(a)) + ctx._2.foreach(a => assume(a)) + val bt = have((r::r.typ) |- ((l*r =:= l.body) === One)) by Restate.from(s1) + val ptlr = have(ProofType(l*r)) + val ptlb = have(ProofType(l.body)) + val bth = have((r::r.typ, l*r :: l.defin.outType, l.body :: l.defin.outType) |- (l*r === l.body)) by Substitution.ApplyRules( + eqCorrect of (HOLSteps.x := l*r, HOLSteps.y := l.body, A := l.defin.outType) + )(bt) + have(Discharge(ptlr)(lastStep)) + have(Discharge(ptlb)(lastStep)) + case _ => + return proof.InvalidProofTactic(s"The term should be of the form (λx. t) x") + } + } + + + // λ(x, t*x) === t + object ETA_PRIM extends ProofTactic { + def apply(using proof: Proof)(x: TypedVar, t: Term): proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + if t.freeVariables.contains(x) then + return proof.InvalidProofTactic(s"Variable $x is free in the term $t") + val lxtx = λ(x, t*x) + val ctx = computeContext(Set(lxtx, t)) + ctx._1.foreach(a => assume(a)) + ctx._2.foreach(a => assume(a)) + have(BETA_PRIM(lxtx*x)) + thenHave((x :: x.typ) ==> (lxtx*x === t*x)) by Restate.from + thenHave(tforall(x, lxtx*x === t*x)) by RightForall + val r1 = have((lxtx:: lxtx.typ, t::lxtx.typ) |- (lxtx === t)) by Tautology.from( + funcUnique of (f := lxtx, g := t, A := x.typ, B := lxtx.defin.outType), + lastStep + ) + val r2 = have((t::lxtx.typ) |- (lxtx === t)) by Cut(have(ProofType(lxtx)), r1) + have((lxtx === t)) by Cut(have(ProofType(t)), r2) + } + } + // λ(x, t*x) =:= t + object ETA extends ProofTactic { + def apply(using proof: Proof)(x: TypedVar, t: Term): proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + if t.freeVariables.contains(x) then + return proof.InvalidProofTactic(s"Variable $x is free in the term $t") + val lxtx = λ(x, t*x) + val ctx = computeContext(Set(lxtx, t)) + ctx._1.foreach(a => assume(a)) + ctx._2.foreach(a => assume(a)) + have(BETA_PRIM(lxtx*x)) + thenHave((x :: x.typ) ==> (lxtx*x === t*x)) by Restate.from + thenHave(tforall(x, lxtx*x === t*x)) by RightForall + val r1 = have((lxtx:: lxtx.typ, t::lxtx.typ) |- ((lxtx =:= t) === One)) by Tautology.from( + funcUnique2 of (f := lxtx, g := t, A := x.typ, B := lxtx.defin.outType), + lastStep + ) + + val r2 = have((t::lxtx.typ) |- ((lxtx =:= t) === One)) by Cut(have(ProofType(lxtx)), r1) + have((lxtx =:= t) === One) by Cut(have(ProofType(t)), r2) + + } + } + + /** + * --------------- + * t |- t + */ + object ASSUME extends ProofTactic { + def apply(using proof: Proof)(t:Term): proof.ProofTacticJudgement = TacticSubproof { + val typ = computeType(t) + if typ == 𝔹 then + have(t |- t) by Restate + else + return proof.InvalidProofTactic(s"Term $t is not a boolean") + } + + } + + + /** + * |- t = u |- t + * ------------------- + * |- u + */ + object _EQ_MP extends ProofTactic { + def apply(using proof: Proof)(eq: proof.Fact, p: proof.Fact): proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + if eq.statement.right.size != 1 then + return proof.InvalidProofTactic(s"The first premise should be of the form (t =:= u) === One") + eq.statement.right.head match + case (=:=(`𝔹`)*t*u) === One => + if p.statement.right.size != 1 then + return proof.InvalidProofTactic(s"The second premise should prove $t but proves ${p.statement.right}") + p.statement.right.head match + case eqOne(`t`) => + val assumptions = eq.statement.left ++ p.statement.left + val vt = typedvar(𝔹) + val hp = have((assumptions + (t :: 𝔹) + (u :: 𝔹)) |- p.statement.right) by Weakening(p) + val h1 = have((assumptions + (t :: 𝔹) + (u :: 𝔹)) |- t === u) by Tautology.from(eqCorrect of (x := t, y := u, A := 𝔹), eq) + val hc = have((assumptions + (t :: 𝔹) + (u :: 𝔹) + (t === u)) |- (u === One)) by RightSubstEq.withParametersSimple(List((t, u)), F.lambda(vt, vt === One))(hp) + val h2 = have((assumptions + (t :: 𝔹) + (u :: 𝔹)) |- (u === One)) by Cut(h1, hc) + val pt = have(ProofType(t)) + val h3 = have(Discharge(pt)(h2)) + val h4 = have(Discharge(have(ProofType(u)))(h3)) + ip.cleanAssumptions + have(Clean.all(h4)) + //println("================================== STRT EQ MP ==================================") + //println(s"eq ${eq.statement}") + //println(s"p ${p.statement}") + //println(s"h2 ${h2.statement}") + //println(s"pt ${pt.statement}") + //println(s"h3 ${h3.statement}") + //println(s"h4 ${h4.statement}") + //println(s"last ${lastStep.statement}") + //println(s"t ++ u \n\t $t \n\t $u") + //println("================================== ENDS EQ MP ==================================") + + case _ => + //println(s"EQ_MP got \n\t${eq.statement}\n\t${p.statement}") + return proof.InvalidProofTactic(s"The second premise should prove $t but proves ${p.statement.right}") + case _ => + return proof.InvalidProofTactic(s"The first premise should be of the form (t =:= u) === One ") + + } + + } + + /** + * |- t = u |- s + * ------------------- + * |- u + * + * where s and t and alpha equivalent. + * + * @see [[_EQ_MP]] + */ + object EQ_MP extends ProofTactic { + def apply(using proof: Proof)(eq: proof.Fact, p: proof.Fact): proof.ProofTacticJudgement = TacticSubproof{ + if eq.statement.right.size != 1 then + return proof.InvalidProofTactic(s"The first premise should be of the form (t =:= u) === One") + eq.statement.right.head match + case (=:=(`𝔹`)*t*u) === One => + p.statement.right.head match + case eqOne(s) => + if s == t then + have(_EQ_MP(eq, p)) + else + val s0 = have(ALPHA_EQUIVALENCE(s, t)) // |- s = t + val s1 = have(_TRANS(s0, eq)) // |- s = u + have(_EQ_MP(s1, p)) + case _ => return proof.InvalidProofTactic(s"Second premise to EQ_MP must be of the form s === One") + + case _ => + return proof.InvalidProofTactic(s"The first premise should be of the form (t =:= u) === One ") + + } + + } + + /** + * A |- p B |- q + * ------------------------- + * A - p, B - q |- p = q + */ + object DEDUCT_ANTISYM_RULE extends ProofTactic { + def apply(using proof: Proof)(t1: proof.Fact, t2: proof.Fact): proof.ProofTacticJudgement = TacticSubproof{ + if t1.statement.right.size != 1 || t2.statement.right.size != 1 then + return proof.InvalidProofTactic(s"The premises should be of the form p === One and q === One") + val left1 = t1.statement.left + val c1 = t1.statement.right.head + val left2 = t2.statement.left + val c2 = t2.statement.right.head + (c1, c2) match + case (eqOne(p), eqOne(q)) => + (left1 - c2).foreach(f => assume(f)) + (left2 - c1).foreach(f => assume(f)) + val qp = have((p :: 𝔹, q :: 𝔹) |- (q === One) ==> (p === One)) by Weakening(t1) + val pq = have((p :: 𝔹, q :: 𝔹) |- (p === One) ==> (q === One)) by Weakening(t2) + val pivot = have((p :: 𝔹, q :: 𝔹) |- (q === One) <=> (p === One)) by RightAnd(pq, qp) + + val h0 = have((p :: 𝔹, q :: 𝔹) |- (p === q)) by Cut(pivot, propExt of (HOLSteps.p -> p, HOLSteps.q -> q)) + val h1 = have((p :: 𝔹, q :: 𝔹) |- (p =:= q === One)) by Tautology.from(h0, eqCorrect of (A -> 𝔹, x -> p, y -> q)) + val h2 = have(Discharge(have(ProofType(p)))(h1)) + have(Discharge(have(ProofType(q)))(h2)) + //println("================================== STRT DEDUCT ==================================") + //println(s"t1 ${t1.statement}") + //println(s"t2 ${t2.statement}") + //println(s"h1 ${h1.statement}") + //println(s"h2 ${h2.statement}") + //println(s"last ${lastStep.statement}") + //println("================================== ENDS DEDUCT ==================================") + + + case _ => + return proof.InvalidProofTactic(s"The premises should be of the form p === One and q === One") + + } + + } + + + + object INST extends ProofTactic { + def apply(using proof: Proof)(inst: Seq[(TypedVar, Term)], prem: proof.Fact): proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + + val h0 = prem.of(inst.map(_ := _): _*) + val h1 = inst.foldLeft(h0: ip.Fact)((acc, p) => + have( Discharge(have( ProofType(p._2) ))(acc) ) + ) + h0.statement.right.head match + case eqOne(r) => + + val def_red_r = have(DEF_RED(r)) // r === r2 + def_red_r.statement.right.head match { + case `r` === r2 => + val s0 = have((h1.statement.left ++ def_red_r.statement.left) |- eqOne(r)) by Weakening(h1) + val s1 = have((h1.statement.left ++ def_red_r.statement.left + (r === r2)) |- eqOne(r2)) by RightSubstEq.withParametersSimple(List(r -> r2), F.lambda(x, eqOne(x)))(s0) + val s2 = have((h1.statement.left ++ def_red_r.statement.left) |- eqOne(r2)) by Cut(def_red_r, s1) + have(Clean.all(s2)) + case fail === _ => + throw new Exception(s"Was expecting an equation with left hand side $r but got $fail") + case _ => + throw new Exception(s"Was expecting an equation as return of DEF_RED but got ${def_red_r.statement.right.head}") + } + + case _ => + throw new Exception(s"Was expecting an r === One but got ${h0.statement.right.head}") + + + } + + + } + + object INST_TYPE extends ProofTactic { + + def apply(using proof: Proof)(x: F.Variable, t:Term, prem: proof.Fact): proof.ProofTacticJudgement = TacticSubproof{ + val r = prem of (x := t) + have(r.statement) by Restate.from(r) + } + + } + + object SYM extends ProofTactic { + def apply(using proof: Proof)(p: proof.Fact): proof.ProofTacticJudgement = TacticSubproof { + val stmt = p.statement + stmt match + case HOLSequent(left, =:=(typ)*s*t) => + // flip + val s0 = have((stmt.left + (s :: typ) + (t :: typ)) |- (s =:= t) === One) by Weakening(p) + val s1 = have((stmt.left + (s :: typ) + (t :: typ)) |- (t =:= s) === One) by Cut(s0, eqSym of (x := s, y := t, A -> typ)) + val s2 = have(Discharge(have(ProofType(s)))(s1)) + val s3 = have(Discharge(have(ProofType(s)))(s2)) + case _ => return proof.InvalidProofTactic(s"Sequent must contain an equality s =:= t to flip.") + } + } + + /** + * Apply equality transitivity, but flip the equalities if necessary. + * + * @see [[_TRANS]] + */ + object _TRANS_SYM extends ProofTactic { + def apply(using proof: Proof)(p1: proof.Fact, p2: proof.Fact): proof.ProofTacticJudgement = + val s1 = p1.statement + val s2 = p2.statement + (s1, s2) match + case (HOLSequent(left1, =:=(aa)*s*ta), HOLSequent(left2, =:=(ab)*tb*u) ) => //equality is too strict + if aa == ab then + // fine as is? + if ta == tb then + _TRANS(p1, p2) + // flip first? + else if s == tb then + _TRANS(have(SYM(p1)), p2) + // flip second? + else if u == ta then + _TRANS(p1, have(SYM(p2))) + // flip both? + else if u == s then + _TRANS(have(SYM(p1)), have(SYM(p2))) + else + proof.InvalidProofTactic(s"Could not construct an instance of transitivity from terms: \n\t$s\n\t$ta\n\t$tb\n\t$u") + else + proof.InvalidProofTactic(s"Types don't agree: $aa and $ab") + + case (HOLSequent(left1, right1), HOLSequent(left2, right2) ) => + proof.InvalidProofTactic(s"The facts should have equalities") + case _ => + s1 match + case HOLSequent(left1, right1) => + proof.InvalidProofTactic(s"The second fact is not parsable as an HOL sequent") + case _ => + proof.InvalidProofTactic(s"The first fact is not parsable as an HOL sequent") + } + + /** + * |- λx. t = λy. t[x := y] if y is not free in t + * + */ + private object _ALPHA_CONV extends ProofTactic { + def apply(using proof: Proof)(t: Term, x: TypedVar, y: TypedVar): proof.ProofTacticJudgement = TacticSubproof { + if !t.freeVariables.contains(y) then + val ld = λ(x, t) + if x == y then + // trivial + have(REFL(ld)) + else + val s0 = have(BETA((ld * x))) // |- (λx. t) x =:= t + val s1 = have(INST(Seq(x -> y), s0)) // |- (λx. t) y =:= t[x := y] + val s2 = have(ABS(y)(s1)) // |- (λy. (λx. t) y) =:= λy. t[x := y] + val s3 = have(ETA(y, ld)) // |- (λy. (λx. t) y) =:= (λx. t) + val s4 = have(_TRANS_SYM(s2, s3)) // |- (λx. t) =:= λy. t[x := y] + else + return proof.InvalidProofTactic(s"Not applicable: the variable $y is free in the term $t.") + } + } + + /** + * Recursively replace every bound x by y in t. + * + * Will not behave nicely on non-HOL terms. + */ + private object _ALPHA_CONV_REC extends ProofTactic { + def apply(using proof: Proof)(t: Term, x: TypedVar, y: TypedVar): proof.ProofTacticJudgement = TacticSubproof { + t match + case abs: Abstraction => + if abs.bound == x then + val b = abs.body + val i0 = have(_ALPHA_CONV(b, x, y)) // |- λx. b = λy. b[x := y] + val i1 = have(_ALPHA_CONV_REC(b, x, y)) // |- b = b.rec(x -> y) + val i2 = have(INST(Seq(x -> y), i1)) // |- b[x := y] = b.rec(x -> y)[x := y] + val i3 = have(ABS(y)(i2)) // |- λy. b[x := y] = λy. b.rec(x -> y)[x := y] + + + have(_TRANS_SYM(i0, i3)) + else + val inner = have(_ALPHA_CONV_REC(abs.body, x, y)) // |- t = t.rec(x -> y) + have(ABS(abs.bound)(inner)) // |- λx. t = λx. t.rec(x -> y) + case af:AppliedFunction => + val func = af.func + val arg = af.arg + val fconv = have(_ALPHA_CONV_REC(func, x, y)) // |- func = func.rec(x -> y) + val aconv = have(_ALPHA_CONV_REC(arg, x, y)) // |- arg = arg.rec(x -> y) + have(MK_COMB(fconv, aconv)) // |- func * arg = func.rec * arg.rec + case _ => + have(REFL(t)) // |- t = t + + val res = lastStep.statement.right.head.asInstanceOf[F.AppliedPredicate].args(0) + } + } + + /** + * Given two terms, if they are alpha-equivalent, produce a proof of it, else fail. + */ + object ALPHA_EQUIVALENCE extends ProofTactic { + + private val name = "alph" + + private def alphaEquivalent(t1: Term, t2: Term): Boolean = + val res = t1 == t2 || { + (t1, t2) match + case (a1: AppliedFunction, a2: AppliedFunction) => + alphaEquivalent(a1.func, a2.func) && alphaEquivalent(a1.arg, a2.arg) + case (a1: (AbstrVar | Abstraction), a2: (AbstrVar | Abstraction)) => + val d1 = a1 match + case a: AbstrVar => a.defin + case a: Abstraction => a.defin + val d2 = a2 match + case a: AbstrVar => a.defin + case a: Abstraction => a.defin + if d1.bound.typ == d2.bound.typ && d1.freeVars.zip(d2.freeVars).forall(_.typ == _.typ) then + val freshBound = TypedVar(F.freshId((d1.freeVariables ++ d2.freeVariables + d1.bound + d2.bound).map(_.id), name), d1.bound.typ) + val freshFrees = + d1.freeVars.foldRight(List.empty[TypedVar]) { + case (v, vars) => + val newVar = TypedVar(F.freshId((d1.freeVariables ++ d2.freeVariables ++ vars + d1.bound + d2.bound).map(_.id), name), v.typ) + newVar :: vars + } + val b1 = d1.body.substitute(d1.freeVars.zip(freshFrees).map(_ := _) :+ (d1.bound := freshBound): _*) + val b2 = d2.body.substitute(d2.freeVars.zip(freshFrees).map(_ := _) :+ (d2.bound := freshBound): _*) + alphaEquivalent(b1, b2) + else false + case _ => + //if t1 != t2 then //println(s"Found ${t1.getClass().getName()} and ${t2.getClass.getName()} and failed") + t1 == t2 + } + //println(s"Checking $t1 and $t2 for alpha eq, it's $res") + res + + + private def alpha(using proof: Proof)(t1: Term, t2: Term): proof.ProofTacticJudgement = TacticSubproof { + (t1, t2) match + case (f1: AppliedFunction, f2: AppliedFunction) => // f * u, g * v + val funs = have(ALPHA_EQUIVALENCE(f1.func, f2.func)) // |- f = g + val args = have(ALPHA_EQUIVALENCE(f1.arg, f2.arg)) // |- u = v + have(MK_COMB(funs, args)) // |- f * u = g * v + case (a1: Abstraction, a2: Abstraction) => // λx. t, λy. s + val fresh = TypedVar(F.freshId((a1.freeVariables ++ a2.freeVariables + a1.bound + a2.bound).map(_.id), name), a1.bound.typ) + val s1 = have(_ALPHA_CONV_REC(t1, a1.bound, fresh)) // |- λx. t = λz. t[x := z] + val s2 = have(_ALPHA_CONV_REC(t2, a2.bound, fresh)) // |- λy. s = λz. s[y := z] + val b1 = a1.body.substitute(a1.bound := fresh) // t[x := z] + val b2 = a2.body.substitute(a2.bound := fresh) // s[y := z] + val inner = have(ALPHA_EQUIVALENCE(b1, b2)) // |- t[x := z] = s[y := z] + val abs = have(ABS(fresh)(inner)) // |- λz. t[x := z] = λz. s[y := z] + val s3 = have(_TRANS(s1, abs)) // |- λx. t = λz. s[y := z] + val s4 = have(SYM(s2)) // |- λz. s[y := z] = λy. s + have(_TRANS(s3, s4)) // |- λx. t = λy. s + case (v1: AbstrVar, v2: Abstraction) => + have(v1 =:= v2) by Sorry + case (v1: Abstraction, v2: AbstrVar) => + have(v1 =:= v2) by Sorry + case (v1: AbstrVar, v2: AbstrVar) => + have(v1 =:= v2) by Sorry + case _ => + //println(s"Found reflexively $t1 and $t2") + have(REFL(t1)) // |- t = t + } + + def apply(using proof: Proof)(t1: Term, t2: Term): proof.ProofTacticJudgement = + if alphaEquivalent(t1, t2) then + alpha(t1, t2) + else proof.InvalidProofTactic(s"Given terms are not alpha equivalent: $t1 and $t2.") + } + + object DEF_RED extends ProofTactic { + def apply(using proof: Proof)(t: Term): proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + t match + case ia: InstAbstraction => // $λ*a*b*c... + val base = ia.base + val insts = ia.insts + ip.cleanAssumptions + val a1 = assume(base.defin.bodyProp) + val eq = insts.foldRight(a1: ip.Fact)((inst, acc) => + val i1 = acc of inst + i1.statement.right.head match + case F.==>(left, right) => // |- inst :: x.typ --- not checking that type of insts match types freevars + + val i2 = have((i1.statement.left + left) |- right) by Restate.from(i1) + val b1 = have(ProofType(inst._2)) + if b1.statement.right.head != left then + throw new Exception(s"Can't instantiate variable $left with term ${inst} of type ${b1.statement.right.head}.") + val b2 = have(Discharge(b1)(i2)) + b2 + case _ => + throw new Exception(s"Was expecting an implication while unfolfing instantiations, but got ${i1.statement.right.head}." + + s"\n The instantiation was $inst. \n The formula was ${a1.statement}." ) + ) + eq.statement.right.head match + case F.forall(x2: F.Variable, F.==>(v is (tp: Term), l === r)) => //ia.repr*insts...*x = ir.repr.body + val x = x2 match + case x: TypedVar => x + case _ => TypedVar(x2.id, tp) + val def_red_r = have(DEF_RED(r)) // r === r2 + def_red_r.statement.right.head match + case `r` === r2 => + val lambdar = λ(x, r2) + val ctx = computeContext(Set(t, lambdar)) + val assump = ctx._1 ++ ctx._2 + val ctxlr = computeContext(Set(lambdar, r2)) + val goal = have((assump + (x::x.typ)) |- (t*x === r)).by.bot + val i0 = have((assump + (x::x.typ)) |- (t*x === r)) by Weakening(eq of x) + val xx = freshVariable[F.Sequent](Seq(i0.statement, def_red_r.statement), "xx") + val i1 = have((assump + (x::x.typ) + (r === r2)) |- (t*x === r2)) by RightSubstEq.withParametersSimple(List((r, r2)), F.lambda(xx, t*x === xx))(i0) + val i2 = have(Discharge(def_red_r)(i1)) + val h = have((assump + (x::x.typ) + (r2 === lambdar*x)) |- t*x === lambdar*x) by RightSubstEq.withParametersSimple(List((r2, lambdar*x)), F.lambda(xx, t*x === xx))(i2) + val pure = have(BETA_PRIM(lambdar*x)) // λ(x, r)*x === r + val h0 = have(Discharge(pure)(h)) + thenHave(assump |- (x::x.typ ==> (t*x === lambdar*x))) by Restate.from + //println(s"DEF RED DATA") + //println(lastStep.statement) + val resSeq = have(assump |- tforall(x, t*x === lambdar*x)).by.bot + //println(resSeq) + //println((resSeq.left ++ resSeq.right).flatMap(_.freeVariables)) + val h1 = thenHave(assump |- tforall(x, t*x === lambdar*x)) by RightForall + val iatyp = x.typ |=> base.defin.outType + val fu = funcUnique of (f := lambdar, g := t, A := x.typ, B := base.defin.outType) + val h2 = have((assump + (t :: iatyp) + (lambdar :: iatyp)) |- (t === lambdar)) by Tautology.from( + fu, + eq, + h1 + ) + val h3 = have(Discharge(have(ProofType(lambdar)))(h2)) + val ptt = have(ProofType(t)) + val h4 = have(Discharge(ptt)(h3)) + + val prop_bodyprop = have(Restate(base.defin |- base.defin.bodyProp)) + val h5 = have(Discharge(prop_bodyprop)(h4)) + + case fail === _ => + throw new Exception(s"Was expecting an equation with left hand side $r but got $fail") + case fail => + throw new Exception(s"Was expecting an equation as return of DEF_RED but got $fail") + case F.forall(x: TypedVar, F.==>(tp, r)) => + throw new Exception("Was expecting something of the form ∀(x, x::T => l === r) but got" + eq.statement) + case F.forall(x: TypedVar, r) => + throw new Exception("Was expecting something of the form ∀(x, x::T => f) but got" + eq.statement) + case F.forall(x: F.Variable, r) => + throw new Exception("Was expecting something of the form ∀(x:TypedVar, x::T => f) but got" + eq.statement) + case r => + throw new Exception(s"Was expecting a formula of the form ∀(x, f) but got $r") + + + case abs: Abstraction => + have(abs === abs) by Restate + case v: TypedVar => + have(v === v) by Restate + case f*u => + val s1 = have(DEF_RED(f)) + val s2 = have(DEF_RED(u)) + //println(s"[DEF RED] Found f $f with definition ${s1.statement}") + //println(s"[DEF RED] Found u $u with definition ${s2.statement}") + (s1.statement.left ++ s2.statement.left).foreach(f => assume(f)) + (s1.statement.right.head, s2.statement.right.head) match + case (`f` === f2, `u` === u2) => + val x = typedvar(computeType(f)) + val y = typedvar(computeType(u)) + val s3 = have(f*u === f*u) by Restate + val k = freshVariable(Set(f*u === f2*u2) ++ lastStep.statement.left, "k") + val s4 = have((f === f2) |- app(f, u) === app(f2, u)) by RightSubstEq.withParametersSimple(List(f -> f2), F.lambda(Seq(k), app(f, u) === app(k, u)))(s3) + val s45 = have((f === f2, u === u2) |- app(f, u) === app(f2, u2)) by RightSubstEq.withParametersSimple(List(u -> u2), F.lambda(Seq(k), app(f, u) === app(f2, k)))(s4) + val s5 = have((u === u2) |- f*u === f2*u2) by Cut(s1, s45) + val s6 = have(f*u === f2*u2) by Cut(s2, s5) + case (fail1, fail2) => + throw new Exception(s"Was expecting two equations with left hand side $f but got $fail1 and with left hand side $u but got $fail2") + case t => + have(t === t) by Restate + } + } + + + object Clean { + def lambda(using proof: Proof)(defin: AbstractionDefinition)(prem: proof.Fact) : proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + ip.cleanAssumptions + val lctx12 = computeContextOfFormulas(Set(defin), Set()) + val lctx = lctx12._1 ++ lctx12._2 + val exdefin = F.exists(defin.reprVar, defin) + + val goal = ((prem.statement.removeAllLeft(defin.args.toSet + defin)) +<< exdefin) + + val s0 = have((prem.statement.removeAllLeft(defin.args.toSet + defin)) +<< defin) by Weakening(prem) + val s2 = have(LeftExists(s0)(goal)) + lctx.foreach(a => + assume(a) + ) + val s1 = defin.elim + val s3 = have(Discharge(s1)(s2)) + } + + def allLambdas(using proof: Proof)(prem: proof.Fact) : proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + val statement = prem.statement + val defins = statement.left.collect{case d: AbstractionDefinition => d} + val candidate = defins.find(d => !(statement.removeAllLeft(d.args.toSet + d)).freeVariables.contains(d.reprVar)) + //println(s"[CLEANING] Found defins ${defins.mkString(" ")}") + //println(s"[CLEANING] Found candidate ${candidate.mkString(" ")}") + + candidate match + case Some(defin: AbstractionDefinition) => + val h = have(Clean.lambda(defin)(prem)) + allLambdas(h) + case None => + have(prem.statement) by Weakening(prem) + } + + def variable(using proof: Proof)(ta: TypeAssignment[Term])(prem: proof.Fact) : proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + ip.cleanAssumptions + val (v, typ) = ta.t match + case v: F.Variable => (v, ta.typ) + case _ => return proof.InvalidProofTactic(s"Can only eliminate type assignment of variables") + + if (prem.statement -<< ta).freeVariables.contains(v) then + return proof.InvalidProofTactic(s"The variable ${v} is used in the premise and it's type assignment can't be eliminated") + + val exv = F.exists(v, ta) + + val p1 = have(TypeNonEmptyProof(ta.typ)) + val p2 = have(LeftExists(prem)(prem.statement -<< ta +<< exv)) + val p3 = have(Discharge(p1)(p2)) + } + + + def allVariables(using proof: Proof)(prem:proof.Fact): proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + val statement = prem.statement + val vars = statement.left.collectFirst[TypeAssignment[Term]]{case f @ ((v:F.Variable) is (typ:Term)) if !(statement -<< f).freeVariables.contains(v) => (v :: typ)} + if vars.nonEmpty then + val h = have(Clean.variable(vars.head)(prem)) + allVariables(h) + else + have(prem.statement) by Weakening(prem) + } + + def typeVar(using proof: Proof)(net: NonEmptyType)(prem: proof.Fact) : proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + val p2 = have(LeftExists(prem)(prem.statement -<< net ++<< nonEmptyTypeExists.statement)) + val p3 = have(Discharge(nonEmptyTypeExists)(p2)) + } + + def allTypeVars(using proof: Proof)(prem:proof.Fact): proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + val statement = prem.statement + val types = statement.left.collectFirst[NonEmptyType]{case net: NonEmptyType => net} + if types.nonEmpty then + val h = have(Clean.typeVar(types.head)(prem)) + allTypeVars(h) + else + have(prem.statement) by Weakening(prem) + } + + def all(using proof: Proof)(prem:proof.Fact): proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + ip.cleanAssumptions + val h1 = have(Clean.allLambdas(prem)) + val h2 = have(Clean.allVariables(h1)) + val h3 = have(Clean.allTypeVars(h2)) + } + } + + + + /* + def HOLSubstType(t:Term, A:F.Variable, u:Term): Term = { + t match + case v: TypedVar if v.typ == A => TypedVar(v.id, u.typ) + } +*/ + + def HOLSubst(t:Term, x:TypedVar, u:Term): Term = { + t match + case v:TypedVar if v.id == x.id && v.typ == x.typ => u + case a:Abstraction => + if a.bound == x then a + else λ(a.bound, HOLSubst(a.body, x, u)) + case a:AppliedFunction => HOLSubst(a.func, x, u)*HOLSubst(a.arg, x, u) + case _ => t + } + +} diff --git a/lisa-sets/src/main/scala/lisa/hol/HOLStepsTests.scala b/lisa-sets/src/main/scala/lisa/hol/HOLStepsTests.scala new file mode 100644 index 000000000..b7cba3638 --- /dev/null +++ b/lisa-sets/src/main/scala/lisa/hol/HOLStepsTests.scala @@ -0,0 +1,304 @@ +package lisa.hol +import lisa.hol.HOLSteps.* +import lisa.prooflib.BasicStepTactic.* +import lisa.prooflib.SimpleDeducedSteps.Restate + +object HOLStepsTests extends lisa.HOL { + + + + // REFL + + + // _TRANS + + println("Starting tests") + val tt = w =:=z + val now = System.currentTimeMillis() + + println("starting test1") + + val test_trans_1 = Theorem((w =:= x, x =:= y, y =:= z) |- (w =:=z)) { + println("enters proof") + val a1 = assume(w =:= x) + val a2 = assume(x =:= y) + val a3 = assume(y =:= z) + val s1 = have(_TRANS(a1, a2)) + have(_TRANS(s1, a3)) + } + + println("starting mk_comb") + // MK_COMB + + val test_mkcomb_1 = Theorem((f =:= g, x =:= y) |- (f*x =:= g*y)) { + val a1 = assume(f =:= g) + val a2 = assume(x =:= y) + have(MK_COMB(a1, a2)) + } + // ABS + + val test_abs_1 = Theorem((y =:= z) |- (λ(x, y) =:= λ(x, z))) { + assume(y =:= z) + have(ABS(x)(lastStep)) + } + + + + val thm_abs_2 = Theorem(λ(x, λ(y, y)) =:= λ(x, λ(z, z))) { + have(λ(y, y) =:= λ(z, z)) by Sorry + have(ABS(x)(lastStep)) + } + + + + val thm_abs_3 = Theorem(λ(x, λ(y, x)) =:= λ(x, λ(z, x))) { + have(λ(y, x) =:= λ(z, x)) by Sorry + have(ABS(x)(lastStep)) + } + + val test_abs_4 = Theorem(λ(x, λ(y, f*x =:= g*(λ(z, y)*x))) =:= λ(x, λ(z, z =:= x))) { + have(λ(y, f*x =:= g*(λ(z, y)*x)) =:= λ(z, z =:= x)) by Sorry + have(ABS(x)(lastStep)) + } + println("starting beta") + // BETA + val test_beta_1 = Theorem( λ(x, x)*x =:= x) { + have(BETA(λ(x, x)*x)) + } + + val test_beta_2 = Theorem( λ(x, x)*x =:= (x)) { + have(BETA(λ(x, x)*x)) + } + + val test_beta_3 = Theorem( λ(x, y)*x =:= (y)) { + have(BETA(λ(x, y)*x)) + } + + val test_beta_4 = Theorem( λ(x, x =:= x)*x =:= (x =:= x)) { + have(BETA(λ(x, x =:= x)*x)) + } + + val test_beta_5 = Theorem( λ(x, x =:= y)*x =:= (x =:= y)) { + have(BETA(λ(x, x =:= y)*x)) + } + + val test_beta_6 = Theorem( λ(x, λ(y, x))*x =:= λ(y, x)) { + have(BETA(λ(x, λ(y, x))*x)) + } + + val test_beta_7 = Theorem( λ(x, λ(y, y))*x =:= λ(y, y)) { + have(BETA(λ(x, λ(y, y))*x)) + } + + val test_beta_8 = Theorem( λ(x, λ(y, x =:= y))*x =:= λ(y, x =:= y)) { + have(BETA(λ(x, λ(y, x =:= y))*x)) + } + + + val test_beta_9 = Theorem( λ(x, λ(y, λ(z, x)))*x =:= λ(y, λ(z, x))) { + have(BETA(λ(x, λ(y, λ(z, x)))*x)) + } + + val test_beta_10 = Theorem( λ(x, λ(y, λ(z, y) =:= λ(w, x)))*x =:= λ(y, λ(z, y) =:= λ(w, x))) { + have(BETA(λ(x, λ(y, λ(z, y) =:= λ(w, x)))*x)) + } + + println("starting eta") + // ETA + + val test_eta_1 = Theorem(λ(x, f*x) =:= f) { + have(ETA(x, f)) + } + val test_eta_prim_1 = Theorem(withCTX(λ(x, f*x) === f)) { + have(ETA_PRIM(x, f)) + } + + val f2 = λ(y, y) + val test_eta_2 = Theorem(λ(x, f2*x) =:= f2) { + have(ETA(x, f2)) + } + val test_eta_prim_2 = Theorem(withCTX(λ(x, f2*x) === f2)) { + have(ETA_PRIM(x, f2)) + } + + val f3 = λ(y, y =:= z) + val test_eta_3 = Theorem(λ(x, f3*x) =:= f3) { + have(ETA(x, f3)) + } + val test_eta_prim_3 = Theorem(withCTX(λ(x, f3*x) === f3)) { + have(ETA_PRIM(x, f3)) + } + + val f4 = λ(y, λ(z, f*y)) + val test_eta_4 = Theorem(λ(x, f4*x) =:= f4) { + have(ETA(x, f4)) + } + val test_eta_prim_4 = Theorem(withCTX(λ(x, f4*x) === f4)) { + have(ETA_PRIM(x, f4)) + } + + val f5 = f2 + val test_eta_5 = Theorem(λ(y, f5*y) =:= f5) { + have(ETA(y, f5)) + } + val test_eta_prim_5 = Theorem(withCTX(λ(y, f5*y) === f5)) { + have(ETA_PRIM(y, f5)) + } + + + // ASSUME + + val b = typedvar(𝔹) + + val test_assume_1 = Theorem(b |- b) { + have(ASSUME(b)) + } + + val test_assume_2 = Theorem((x =:= x) |- (x =:= x)) { + have(ASSUME(x =:= x)) + } + + val test_assume_3 = Theorem( (λ(x, y) =:= λ(x, y)) |- (λ(x, y) =:= λ(x, y)) ){ + have(ASSUME(λ(x, y) =:= λ(x, y))) + } + + val expr = λ(f, λ(x, f*x =:= f*(h*v)))*λ(y, f*y)*y + val test_assume_4 = Theorem( expr |- expr ){ + have(ASSUME(expr)) + } + + + val (a1, a2) = (p, q) + val test_eqmp_1 = Theorem((a1 =:= a2, a1) |- a2) { + val s1 = assume(p =:= q) + val s2 = assume(p) + have(_EQ_MP(s1, s2)) + } + + val (a3, a4) = (λ(x, p) =:= λ(x, p), λ(p, q)*p) + val test_eqmp_2 = Theorem((a3 =:= a4, a3) |- a4) { + val s1 = assume(a3 =:= a4) + val s2 = assume(a3) + have(_EQ_MP(s1, s2)) + } + + val test_eqmp_3 = Theorem(λ(p, p)*p |- p ) { + val s1 = have(BETA(λ(p, p)*p)) + val s2 = assume(λ(p, p)*p) + have(_EQ_MP(s1, s2)) + } + + + val test_eqmp_4 = Theorem( p ) { + val s1 = have(BETA(λ(q, p)*q)) + val s2 = have(λ(q, p)*q) by Sorry + have(_EQ_MP(s1, s2)) + } + + + + val test_deductantisymrule_1 = Theorem(withCTX(((p === One) ==> (q === One), (q === One) ==> (p === One)) |- ((p =:= q) === One))){ + assume((p === One) ==> (q === One)) + assume((q === One) ==> (p === One)) + val s1 = have(q |- p) by Restate + val s2 = have(p |- q) by Restate + have(DEDUCT_ANTISYM_RULE(s1, s2)) + } + + println("start inst tests") + + val test_inst_1 = Theorem(q){ + have(p) by Sorry + have(INST(Seq((p, q)), lastStep)) + } + + val test_inst_2 = Theorem(q) { + have(q) by Sorry + have(INST(Seq((p, p=:=p)), lastStep)) + } + + val test_inst_3 = Theorem(p =:= p){ + have(p =:= q) by Sorry + have(INST(Seq((q, p)), lastStep)) + } + + val test_inst_4 = Theorem(p =:= q) { + have(p) by Sorry + have(INST(Seq((p, p=:=q)), lastStep)) + } + + val test_inst_5 = Theorem(λ(x, y)*z =:= z){ + have(λ(x, y)*w =:= w) by Sorry + have(INST(Seq((w, z)), lastStep)) + } + + val test_inst_6 = Theorem(λ(x, y)*z =:= y){ + have(BETA(λ(x, y)*x)) + have(INST(Seq((x, z)), lastStep)) + } + + val test_inst_7 = Theorem(λ(x, x)*z =:= z){ + have(λ(x, x)*x =:= x) by Sorry + have(INST(Seq((x, z)), lastStep)) + } + + val test_inst_8 = Theorem(λ(x, x =:= y)*z =:= (z =:= y)){ + have(BETA(λ(x, x =:= y)*x)) + have(INST(Seq((x, z)), lastStep)) + } + + val test_inst_9 = Theorem(λ(x, λ(y, x))*z =:= λ(y, z)){ + have(BETA(λ(x, λ(y, x))*x)) + have(INST(Seq((x, z)), lastStep)) + } + + val test_inst_10 = Theorem(λ(x, λ(y, y) =:= λ(y, x))*z =:= (λ(y, y) =:= λ(y, z))){ + have(BETA(λ(x, λ(y, y) =:= λ(y, x))*x)) + have(INST(Seq((x, z)), lastStep)) + } + + val test_inst_11 = Theorem(λ(x, λ(y, λ(z, x)))*w =:= λ(y, λ(z, w))){ + have(BETA(λ(x, λ(y, λ(z, x)))*x)) + have(INST(Seq((x, w)), lastStep)) + } + + val test_inst_12 = Theorem(λ(p, q)*p){ + have(λ(p, r)*p) by Sorry + have(INST(Seq((r, q)), lastStep)) + } + + val test_inst_13 = Theorem(λ(x, λ(x, y)*x)*y =:= y){ + val s1 = have(BETA(λ(x, λ(x, y)*x)*x)) + val s2 = have(INST(Seq((x, y)), s1)) // λ(x, λ(x, y)*x)*y === λ(x, y)*y + val s3 = have(BETA(λ(x, y)*x)) // λ(x, y)*x =:= y + val s4 = have(INST(Seq((x, y)), s3)) // λ(x, y)*y =:= y + have(_TRANS(s2, s4)) + } + + + val test_inst_14 = Theorem(λ(x, f*z) =:= λ(x, f*z)){ + val s0 = have(REFL(λ(x, v))) + val s1 = have(INST(Seq((v, f*z)), s0)) + val s2 = have(REFL(λ(x, f*z) )) + have(_TRANS(s1, s2)) + + } + + + + // Those don't hold because they require alpha equivalence to conclude the proof. +/* + println("Starting test 15") + val test_inst_15 = Theorem(λ(q, p)*p){ + have(λ(p, r)*p) by Sorry + have(INST(Seq((r, p)), lastStep)) + } + + println("Starting test 16") + val test_inst_16 = Theorem(λ(x, λ(y, x))*y =:= λ(z, y)){ + have(BETA(λ(x, λ(y, x))*x)) + have(INST(Seq((x, y)), lastStep)) + } +*/ + +} diff --git a/lisa-sets/src/main/scala/lisa/hol/HOLTest.scala b/lisa-sets/src/main/scala/lisa/hol/HOLTest.scala new file mode 100644 index 000000000..60f994eb6 --- /dev/null +++ b/lisa-sets/src/main/scala/lisa/hol/HOLTest.scala @@ -0,0 +1,49 @@ +package lisa.hol + +object HOLTest extends lisa.HOL{ + + val x = typedvar(𝔹) + val y = typedvar(𝔹) + val f = typedvar(𝔹 |=> 𝔹) + val g = typedvar(𝔹 |=> (𝔹 |=> 𝔹)) + val h = typedvar((𝔹 |=> 𝔹) |=> 𝔹) + + output("------Expression 1------") + val expr1 = g*(x)*(f*(y)) + output("expr1: " + expr1) + output("expr1 type: " + computeType(expr1)) + + val typecheckTest = TypingTheorem(expr1 :: 𝔹) + + + output("------Expression 2------") + val expr2 = g*(x)*(λ(x, f*(x))*(y)) + output("expr2: " + expr2) + output("expr2 type: " + computeType(expr2)) + + val typecheckTest2 = TypingTheorem(expr2 :: 𝔹) + + + output("------Expression 3------") + val expr3 = x =:= y + output("expr3: " + expr3) + output("expr3 type: " + computeType(expr3)) + + val typecheckTest3 = TypingTheorem(expr3 :: 𝔹 ) + + + output("------Expression 4------") + val expr4 = (g*(x)) =:= λ(x, f*(x)) + output("expr4: " + expr4) + output("expr4 type: " + computeType(expr4)) + + val typecheckTest4 = TypingTheorem(expr4 :: 𝔹 ) + + output("------Expression 5------") + val expr5 = λ(h, λ(f, f*(x)) =:= h) + output("expr5: " + expr5) + output("expr5 type: " + computeType(expr5)) + + val typecheckTest5 = TypingTheorem(expr5 :: (((𝔹 |=> 𝔹) |=> 𝔹) |=> 𝔹) ) + +} diff --git a/lisa-sets/src/main/scala/lisa/hol/VarsAndFunctions.scala b/lisa-sets/src/main/scala/lisa/hol/VarsAndFunctions.scala new file mode 100644 index 000000000..65a160935 --- /dev/null +++ b/lisa-sets/src/main/scala/lisa/hol/VarsAndFunctions.scala @@ -0,0 +1,646 @@ + +package lisa.hol +import lisa.maths.settheory.types.TypeSystem.{`*` => _, *} +import lisa.maths.settheory.types.TypeLib +import lisa.maths.settheory.types.TypeLib.{𝔹 => _, *} +import TypeChecker.* +import lisa.maths.settheory.SetTheory.{pair, ∅} + +import lisa.fol.FOL as F +import lisa.fol.FOL.{Identifier, Term} +import lisa.prooflib.ProofTacticLib.ProofTactic +import lisa.SetTheoryLibrary +import lisa.hol.VarsAndFunctions.HOLSequent.toHOLSequent +import lisa.maths.settheory.SetTheory.cartesianProduct +import lisa.maths.settheory.SetTheory.secondInPair +import lisa.fol.FOLHelpers.freshVariable +import lisa.utils.unification.UnificationUtils.matchTerm + + +object VarsAndFunctions { + + private given lisa.prooflib.OutputManager = lisa.maths.settheory.SetTheory.om + + def main(args: Array[String]): Unit = { + val x = typedvar(𝔹) + val testTerm = Abstraction(x, x) + + println(testTerm) + } + + type Type = Term + + def computeContext(terms: Set[Term]) = computeContextKnown(terms, Set.empty) + + def computeContextKnown(terms: Set[Term], known: Set[Variable]): (Set[VarTypeAssignment], Set[AbstractionDefinition]) = + val frees = terms.flatMap(_.freeVariables) -- known + val (r1, r2) = frees.foldLeft((List.empty[VarTypeAssignment], List.empty[AbstractionDefinition])) { + case ((acc1, acc2), a: AbstrVar) => + (acc1, a.defin :: acc2) + case ((acc1, acc2), v: TypedVar) => + ((v is v.typ).asInstanceOf[VarTypeAssignment] :: acc1, acc2) + case ((acc1, acc2), v) => + (acc1, acc2) + } + val rec = if r2.isEmpty then (Set(), Set()) else computeContextOfFormulas(r2.toSet, frees) + (r1.toSet, r2.toSet ++ rec._2) + + def computeContextOfFormulas(formulas: Set[Formula], known: Set[Variable] = Set()): (Set[VarTypeAssignment], Set[AbstractionDefinition]) = + val vars = formulas.flatMap(_.freeVariables) -- known + computeContextKnown(vars.toSet, Set.empty) + + private def HOLSeqToFOLSeq(left: Set[Term], right: Term): (Set[VarTypeAssignment], Set[AbstractionDefinition]) = { + computeContext(left + right) + } + + class HOLSequent( + val premises: Set[Term], + val conclusion: Term, + val varTypes: Set[VarTypeAssignment], + val abstrs: Set[AbstractionDefinition] + ) extends F.Sequent(premises.map(_ === One) ++ varTypes ++ abstrs, Set(conclusion === One)) { + + infix def +<<(t: Term): HOLSequent = HOLSequent(this.premises + t, conclusion) + infix def -<<(t: Term): HOLSequent = HOLSequent(this.premises - t, conclusion) + + override infix def +<<(f: Formula): F.Sequent = + f match + case ===(t, One) => +<<(t) + case ===(One, t) => +<<(t) + case _ => super.+<<(f) + override infix def -<<(f: Formula): F.Sequent = + f match + case ===(t, One) => -<<(t) + case ===(One, t) => -<<(t) + case _ => super.-<<(f) + + infix def ++<<(s1: HOLSequent): HOLSequent = HOLSequent(this.premises ++ s1.premises, conclusion) + infix def --<<(s1: HOLSequent): HOLSequent = HOLSequent(this.premises -- s1.premises, conclusion) + + override infix def ++<<(s1: F.Sequent): F.Sequent = + s1 match + case s1: HOLSequent => ++<<(s1) + case s1: F.Sequent => super.++<<(s1) + override infix def --<<(s1: F.Sequent): F.Sequent = + s1 match + case s1: HOLSequent => --<<(s1) + case s1: F.Sequent => super.--<<(s1) + + } + + + + import F.{given} + + class ConstantTypeTerm(id: Identifier, val nonEmptyThm: JUSTIFICATION) extends Constant(id) + + private given TypeLib.library.type = TypeLib.library + val boolNonEmpty = Theorem(exists(x, in(x, TypeLib.𝔹))) { + have(thesis) by RightExists(One.justif) + } + + val G = function[1] + val λ = variable + //val functionsRelativization = Axiom(∀(x, (x ∈ A) ==> (G(x) ∈ B)) ==> ∃(λ, (λ ∈ (A |=> B)) /\ ∀(x, (λ*x) === G(x))) ) + + + val functionalExtentionality = Axiom({ + val f = typedvar(A |=> B) + val g = typedvar(A |=> B) + val x = typedvar(A) + ((f :: (A |=> B)) /\ (g :: (A |=> B)) /\ tforall(x, f*x === g*x)) ==> (f === g) + }) + + val T = variable + val nonEmptyTypeExists = Theorem(exists(T, exists(x, in(x, T)))) { + have(thesis) by RightExists(boolNonEmpty) + } + val 𝔹 = ConstantTypeTerm("𝔹", boolNonEmpty) + + + val =:= : TypedConstantFunctional[1] ={ + val =:= = F.ConstantFunctionLabel.infix("=:=", 1) + addSymbol(=:=) + val typ = (A |=> (A |=> 𝔹)) + val typing_of_eq = Axiom(F.forall(A, =:=(A) :: typ)) + TypedConstantFunctional[1]("=:=", 1, FunctionalClass(Seq(any), Seq(A), (A |=> (A |=> 𝔹)), 1), typing_of_eq) + } + lazy val eqDefin = { + val x = typedvar(A) + val y = typedvar(A) + val eqDefin = Axiom(((x::A) /\ (y::A)) ==> ((x =:= y)===One) <=> (x===y)) + eqDefin + } + + val holeq : TypedConstantFunctional[1] = VarsAndFunctions.=:= + + object eqOne { + def unapply(f: Formula): Option[Term] = f match { + case ===(t, One) => Some(t) + case ===(One, t) => Some(t) + case _ => None + } + + def apply(t: Term): Formula = t === One + } + + given Conversion[Term, F.Formula] = t => eqOne(t) + + extension (t1:Term) { + def =:=(t2:Term): Term = + val A = computeType(t1) + if (A == computeType(t2)) + holeq.applySeq(Seq(A))*(t1)*(t2) + else + throw new TypingException("in expression " + t1 + " =:= " + t2 + " the types " + A + "of left-hand side and " + computeType(t2) + " of right-hand side do not match.") + def equalityOfType(A:Term) (t2:Term): Term = holeq.applySeq(Seq(A))*(t1)*(t2) //compute A with computeType, possibly. + } + + object HOLSequent { + def apply(premises: Set[Term], conclusion: Term): HOLSequent = { + val (valTypes, abstr) = HOLSeqToFOLSeq(premises, conclusion) + new HOLSequent(premises, conclusion, valTypes, abstr) + } + + def toHOLSequent(s: F.Sequent): HOLSequent = + if s.isInstanceOf[HOLSequent] then + return s.asInstanceOf[HOLSequent] + if s.right.size != 1 then + throw new IllegalArgumentException("Sequent must have exactly one conclusion.") + val r = s.right.head + r match + case eqOne(t) => + var vartypes = List.empty[VarTypeAssignment] + var abstr = List.empty[AbstractionDefinition] + var prems = Set.empty[Term] + s.left.foreach { + case v: VarTypeAssignment => vartypes = v :: vartypes + case a: AbstractionDefinition => abstr = a :: abstr + case eqOne(t) => prems = prems + t + case _ => throw new IllegalArgumentException("Premises must be of the form t === One, be a type assignment or an abstraction definition.") + } + new HOLSequent(prems.toSet, t, vartypes.toSet, abstr.toSet) + case _ => + throw new IllegalArgumentException("Conclusion must be of the form t === One.") + + + + def unapply(s: F.Sequent): Option[(Set[Term], Term)] = + if s.isInstanceOf[HOLSequent] then + val s1 = s.asInstanceOf[HOLSequent] + Some((s1.premises, s1.conclusion)) + else + try { + val s1 = toHOLSequent(s) + Some((s1.premises, s1.conclusion)) + } + catch + case e: IllegalArgumentException => + println(e.getMessage()) + return None + + + } + + + def TypingTheorem(using om: lisa.prooflib.OutputManager, name: sourcecode.FullName)(assignment: TypeAssignment[Type]): THM = + val (l1, l2) = HOLSeqToFOLSeq(Set.empty, assignment.t) + Theorem(using om, name)(F.Sequent(l1 ++ l2, Set(assignment.t is assignment.typ))) { + have(thesis) by TypeChecker.prove + } + + extension (t:Term) { + def * (t2:Term): Term = HOLApplication(t, t2) + } + + object * {def unapply(t: Term): Option[(Term, Term)] = t match { + case AppliedFunction(f, a) => Some((f, a)) + case app(f, a) => Some((f, a)) + case _ => None + }} + + /////////////////////////////////////// + /////////// Typed Variables /////////// + /////////////////////////////////////// + + class TypedForall( val v: Variable, val prop: Formula ) extends BinderFormula(forall, v, v match { + case v: TypedVar => (v is v.typ) ==> prop + case _ => prop + } + ) { + override def toString = + val typeStr = v match + case v: TypedVar => s" : ${v.typ}" + case _ => "" + s"∀$v$typeStr. $prop" + } + + + def tforall(v: TypedVar, prop: Formula): TypedForall = TypedForall(v, prop) + + var counter: Int = 0 + + type VarTypeAssignment = Formula & TypeAssignment[Type] {val t:Variable} + + + def nextId: Identifier = { + counter += 1 + Identifier("$λ", counter) + } + + sealed trait TypedHOLTerm { + this : Term => + val typ: Type + def asTerm: Term & TypedHOLTerm = this + var typThm: Option[JUSTIFICATION] = None // conditional proof of this :: typ + } + + class PolymorphicConstant[N <: Arity](label: TypedConstantFunctional[N], args: Term**N) extends AppliedFunctional(label, args.toSeq) with TypedHOLTerm { + val typ = { + val subst = (label.typ.args zip args.toSeq).map((v, a) => (v := a)) + label.typ.out.asInstanceOf[Term].substitute(subst: _*) + } + + } + class HOLConstant(id: Identifier, override val typ: Term, val thm: JUSTIFICATION) extends TypedConstant[Term](id, typ, thm) with TypedHOLTerm { + typThm = Some(thm) + } + + type HOLTerm = HOLApplication | TypedVar | Abstraction | TypedConstant[Term] | PolymorphicConstant[?] + + var i = 0 + extension (ht: HOLTerm) + def typ = ht match + case ht: TypedHOLTerm => ht.typ + case tc: TypedConstant[?] => tc.typ.asInstanceOf[Term] + + def getTypThmOrElseUpdate(computeProof: Proof ?=> Unit): JUSTIFICATION = { + ht match + case ht: TypedHOLTerm => + ht.typThm match + case Some(thm) => thm + case None => + val name = "¢typ_hol_" + i + i += 1 + val ctx = computeContext(Set(ht.asTerm)) + val newthm = THM( (ctx._1 ++ ctx._2) |- (ht.asTerm::ht.typ), name, summon[sourcecode.Line].value, summon[sourcecode.File].value, InternalStatement)(pr ?=> computeProof(using pr)) + ht.typThm = Some(newthm) + newthm + case tc: TypedConstant[?] => tc.justif + + + } + + class HOLApplication(func: Term, arg: Term) extends AppliedFunction(func, arg) with TypedHOLTerm { + val typ = computeType(func) match { + case inType |=> outType => + if computeType(arg) == inType then outType + else + throw new IllegalArgumentException("Argument " + arg + " of function " + func + " has type " + computeType(arg) + " instead of expected " + inType + ".") + case funcType => throw new IllegalArgumentException("Function " + func + " expected to have function type A |=> B, but has type " + funcType + ". ") + } + + } + + class TypedVar( id: Identifier, val typ: Type ) extends Variable(id) with TypedHOLTerm { + override def substituteUnsafe(map: Map[SchematicLabel[?], LisaObject[?]]): Term = + + if map.contains(this) then map(this).asInstanceOf[Term] + else + val typ2 = typ.substituteUnsafe(map) + if typ2 == typ then this + else new TypedVar(id, typ2) + + def toStringFull = s"(${id.name}: $typ)" + + def instType(map: Map[SchematicLabel[?], LisaObject[?]]): TypedVar = new TypedVar(id, typ.substituteUnsafe(map)) + + } + + def typedvar(using name: sourcecode.Name)(typ: Type): TypedVar = new TypedVar(Identifier(name.value), typ) + + + /////////////////////////////////////// + ///////// Lambda Abstractions ///////// + /////////////////////////////////////// + + + class AbstrVar( id: Identifier, val defin:AbstractionDefinition ) extends TypedVar(id, defin.typ){ + } + + + + trait Abstraction extends TypedHOLTerm{ + this : Term => + override def asTerm: Abstraction & Term = this + + val bound: TypedVar + val body: Term + val repr: AbstrVar + val freeVars: Seq[TypedVar] + val defin: AbstractionDefinition + + + val origin: Term + + val typ: Type + + + override def toString = s"${repr.id}($bound. $body)" + + + private lazy val t = this * bound + lazy val betaName = "¢beta_" + repr.id + // import HOLSteps.{=:= => _, *, given} + + + lazy val BETA = THM( t =:= body, betaName, summon[sourcecode.Line].value, summon[sourcecode.File].value, InternalStatement) { + val context = VarsAndFunctions.computeContext(Set(t, body)) + assume((context._1 ++ context._2).toSeq: _*) + val outType = defin.outType + val pro = have(defin.bodyProp |- defin.bodyProp) by Restate + freeVars.reverse.foreach(v => + have(lastStep.statement.right.head.asInstanceOf[TypedForall].prop) by Weakening(lastStep of v) + ) + val aftFreeVars = lastStep + val h = have((bound::bound.typ) |- (t === body)) by Weakening(aftFreeVars of bound) + val h2 = have((bound::bound.typ, t::outType, body::outType) |- ((t =:= body) === One) ) by Substitution.ApplyRules(HOLSteps.eqCorrect of (x := t, y := body, A := outType))(h) + val h3 = have(ProofType(body)) + val h4 = have(((bound::bound.typ, t::outType) |- ((t =:= body) === One)) ++ acc*v), freeVars.last) with Abstraction { + + //override def toString = s"(λ$bound. $body)" + val origin = AppliedFunction(freeVars.init.foldLeft(repr: Term)((acc, v) => acc*v), freeVars.last) + val typ = bound.typ |=> defin.outType + override def substituteUnsafe(map: Map[F.SchematicLabel[?], F.LisaObject[?]]): AppliedFunction = + if map.contains(repr) then super.substituteUnsafe(map) + else + val r = InstAbstraction(this, freeVars.map(v => map.getOrElse(v, v)).asInstanceOf) + val exp = super.substituteUnsafe(map) + if !(r == exp) then + println("r: " + r) + println("exp: " + exp) + r + } + + class InstAbstraction( + val base: Abstraction, + val insts: List[Term] + ) extends AppliedFunction(insts.init.foldLeft(base.repr: Term)((acc, v) => acc @@ v), insts.last) with TypedHOLTerm { + val typ = base.typ + override def toString(): String = "I"+super.toString() + + } + + + object Abstraction { + + val cache = collection.mutable.Map.empty[(TypedVar, Term), Abstraction & Term] + def apply(bound: TypedVar, body: Term): Abstraction & Term = { + cache.getOrElseUpdate((bound, body), { + val freeVars: Seq[TypedVar] = (body.freeVariables - bound).toSeq.sortBy(_.id.name).collect { + case v: TypedVar if !v.isInstanceOf[AbstrVar] => v + } + val repr = Variable(nextId) + val inner = tforall(bound, + (freeVars.foldLeft[Term](repr) { (acc, v) => + acc @@ v + } @@ bound) === body + ) + val bodyProp = freeVars.foldLeft[Formula](inner) { (acc, v) => + tforall(v, acc) + } + val outType = computeType(body) + val defin = new AbstractionDefinition(repr, bound, body, freeVars, outType, bodyProp) + if freeVars.isEmpty then new AbstractionClosureWithoutFreeVars(repr.id, bound, body, defin) + else new AbstractionClosureWithFreeVars(AbstrVar(repr.id, defin), bound, body, freeVars, defin) + }.asTerm) + } + } + def λ(bound: TypedVar, body: Term) = Abstraction(bound, body) + + class AbstractionDefinition( + val reprVar: Variable, + val bound: TypedVar, + val body: Term, + val freeVars: Seq[TypedVar], + val outType: Type, + val bodyProp: Formula + ) extends AppliedConnector(And, Seq(reprVar is freeVars.foldRight(bound.typ |=> outType)((v, acc) => v.typ |=> acc), bodyProp)) { + val typ = freeVars.foldRight(bound.typ |=> outType)((v, acc) => v.typ |=> acc) + + override def substituteUnsafe(map: Map[F.SchematicLabel[?], F.LisaObject[?]]): Formula = + if map.contains(reprVar) then super.substituteUnsafe(map) + else + val newMap = map - bound -- freeVars + AbstractionDefinition( + reprVar, + bound.instType(map - bound), + body.substituteUnsafe(newMap), + freeVars.map(_.instType(newMap)), + outType.substituteUnsafe(newMap), + bodyProp.substituteUnsafe(newMap) + ) + + + import HOLSteps.{=:= => _, *, given} + lazy val elimName = "¢elim_" + reprVar.id + lazy val elim = Axiom(exists(reprVar, this)) + + + + } + + var j: Int = 0 + + + + + object ProofType extends ProofTactic { + def apply2(using proof: SetTheoryLibrary.Proof)(t:Term): proof.ProofTacticJudgement = + val context = computeContext(Set(t)) + TypeChecker.typecheck(context._1.toSeq ++ context._2.toSet, t, None) + + + + def apply(using proof: SetTheoryLibrary.Proof)(t:Term): proof.ProofTacticJudgement = TacticSubproof { + given SetTheoryLibrary.type = SetTheoryLibrary + val r = if true then apply2(t) else t match + case t: TypedHOLTerm => + //println("Good: Caching for " + t) + val r = t.asInstanceOf[HOLTerm].getTypThmOrElseUpdate { + have(apply2(t.asTerm)) + } + lisa.prooflib.SimpleDeducedSteps.Restate.from(r)(r.statement) + case tc: TypedConstant[?] => + //println("Good: Caching for " + t) + lisa.prooflib.SimpleDeducedSteps.Restate.from(tc.justif)(tc.justif.statement) + case _ => + println("Warning: No caching for " + t) + println("it has class " + t.getClass()) + apply2(t) + //val rt = apply2(t) + + val r1 = have(r) + val r2 = have(r1.statement) by lisa.prooflib.BasicStepTactic.Weakening(r1) + } + + + } + + + + + + + var debug = false + def computeType(t:Term): Type = + val r = { + t match + case t: TypedVar => + t.typ + case t: TypedConstant[?] => + t.typ match + case t: Term => t + case _ => throw new IllegalArgumentException("computeTypes only support subterms typed by terms, not untyped or typed by classes.") + case t: AppliedFunction => + val funcType = computeType(t.func) + funcType match + case inType |=> outType => + if computeType(t.arg) == inType then outType + else + throw new IllegalArgumentException("Argument " + t.arg + " of function " + t.func + " has type " + computeType(t.arg) + " instead of expected " + inType + ".") + case funcType => throw new IllegalArgumentException("Function " + t.func + " expected to have function type A |=> B, but has type " + funcType + ". ") + case AppliedFunctional(label, args) => + label match + case label: TypedConstantFunctional[?] => + val labelType = label.typ + if args.zip(labelType.in).forall((arg, inType) => + (inType == any) || { + val argType = computeType(arg) + K.isSame((arg is inType).asFormula.underlying, (arg is argType).asFormula.underlying) + } + ) then + val subst = (labelType.args zip args).map((v, a) => (v := a)) + labelType.out match { + case t: Term => t.substitute(subst: _*) + case f: (Term**1 |-> Formula) @unchecked => throw new IllegalArgumentException("computeTypes only support subterms typed by terms, not untyped or typed by classes.") + } + else + val argsTypes = args.map(arg => + try computeType(arg) + catch + case e: IllegalArgumentException => "?" + computeType + ) + throw new IllegalArgumentException("Function " + label + " has type " + labelType + " but was applied to arguments " + args + " of types " + argsTypes + ".") + case _ => + throw new IllegalArgumentException("computeTypes only support subterms typed by terms, not untyped or typed by classes.") + case _ => + throw new IllegalArgumentException("computeTypes only support fully typed terms. " + t + " is not fully typed.") + } + r + + + + object TypeNonEmptyProof extends ProofTactic { + val A = variable + val B = variable + val nonEmptyFuncSpace = Axiom(exists(x, in(x, B)) ==> exists(x, in(x, (A |=> B)))) + + def apply(using proof: Proof)(typ: Term): proof.ProofTacticJudgement = TacticSubproof{ ip ?=> + typ match { + case ctt: ConstantTypeTerm => have(ctt.nonEmptyThm) + case v: Variable => + val x = freshVariable(Set(v), "x") + have(exists(x, in(x, v)) |- exists(x, in(x, v))) + case a |=> b => + val x = freshVariable(Set(a, b), "x") + val s1 = have(TypeNonEmptyProof(b)) + have(exists(x, in(x, a |=> b))) by Tautology.from(s1, nonEmptyFuncSpace of (A := a, B := b)) + } + } + } + + + + + class NonEmptyType(val x: Variable, val typ: Term) extends BinderFormula(exists, x, in(x, typ)) { + override def toString = s"∃${variable}. ${typ}" + } + object NonEmptyType { + def apply(typ: Term): NonEmptyType = + val x = freshVariable(Set(typ), "x") + new NonEmptyType(x, typ) + def unapply(t: NonEmptyType): Some[(Variable, Term)] = Some(t.x, t.typ) + } + + // Sequent Syntax + + trait TermSetConverter[T] { + def apply(t: T): Set[Term] + } + + given TermSetConverter[Unit] with { + override def apply(u: Unit): Set[Term] = Set.empty + } + + given TermSetConverter[EmptyTuple] with { + override def apply(t: EmptyTuple): Set[Term] = Set.empty + } + + given [H <: Term, T <: Tuple](using c: TermSetConverter[T]): TermSetConverter[H *: T] with { + override def apply(t: H *: T): Set[Term] = c.apply(t.tail) + t.head + } + + given term_to_set[T <: Term]: TermSetConverter[T] with { + override def apply(f: T): Set[Term] = Set(f) + } + + given term_iterable_to_set[T <: Term, I <: Iterable[T]]: TermSetConverter[I] with { + override def apply(s: I): Set[Term] = s.toSet + } + + private def any2set[A, T <: A](any: T)(using c: TermSetConverter[T]): Set[Term] = c.apply(any) + + extension [A, T1 <: A](left: T1)(using TermSetConverter[T1]) { + infix def |-(right: Term): HOLSequent = HOLSequent(any2set(left), right) + infix def ⊢(right: Term): HOLSequent = HOLSequent(any2set(left), right) + } + + given Conversion[Term, HOLSequent] = HOLSequent(Set(), _) + +} diff --git a/lisa-sets/src/main/scala/lisa/maths/settheory/types/TypeSystem.scala b/lisa-sets/src/main/scala/lisa/maths/settheory/types/TypeSystem.scala new file mode 100644 index 000000000..ca496cf5f --- /dev/null +++ b/lisa-sets/src/main/scala/lisa/maths/settheory/types/TypeSystem.scala @@ -0,0 +1,536 @@ +package lisa.maths.settheory.types + +import lisa.prooflib.ProofTacticLib.* +import lisa.fol.FOL +import lisa.automation.Tautology +import lisa.fol.FOL.{*, given} +import lisa.prooflib.BasicStepTactic.* +import lisa.prooflib.SimpleDeducedSteps.* +import lisa.SetTheoryLibrary.{given, *} +import lisa.SetTheoryLibrary +import lisa.kernel.proof.SequentCalculus.SCProofStep +import lisa.maths.settheory.SetTheory.functional +import lisa.prooflib.OutputManager +import lisa.maths.settheory.SetTheory.{singleton, app} +import lisa.hol.HOLSteps.p + +object TypeLib extends lisa.Main { + + import TypeSystem.* + + val |=> : ConstantFunctionLabel[2] = ConstantFunctionLabel.infix("|=>", 2) + private inline def temp = |=> + extension (t:Term) { + def |=>(o:Term): Term = TypeLib.temp(t, o) + } + val app: ConstantFunctionLabel[2] = lisa.maths.settheory.SetTheory.app + addSymbol(|=>) + + val f = variable + val x = variable + val y = variable + val z = variable + val A = variable + val B = variable + val F = function[1] + val funcspaceAxiom = Axiom(f ∈ (A |=> B) ==> (x ∈ A ==> app(f, x) ∈ B)) + val any = DEF(x) --> top + + + // A |=> B is the set of functions from A to B + // C |> D is the functional class of functionals from the class C to the class D + // F is C |> D desugars into ∀(x, (x is C) => (F(x) is D)) + + val testTheorem = Theorem((x is A, f is (A |=> B), F is (A |=> B) |> (A |=> B) ) |- (F(f)*(x) is B)) { + have(thesis) by TypeChecker.prove + } + + val 𝔹 = DEF() --> unorderedPair(∅, singleton(∅)) + + val empty_in_B = Theorem( (∅ :: 𝔹) ) { + have( ∅ :: unorderedPair(∅, singleton(∅))) by Tautology.from(pairAxiom of (z := ∅, x := ∅, y := singleton(∅))) + thenHave(thesis ) by Substitution.ApplyRules(𝔹.shortDefinition) + } + + val sing_empty_in_B = Theorem( (singleton(∅) :: 𝔹) ) { + have( singleton(∅) :: unorderedPair(∅, singleton(∅))) by Tautology.from(pairAxiom of (z := singleton(∅), x := ∅, y := singleton(∅))) + thenHave(thesis ) by Substitution.ApplyRules(𝔹.shortDefinition) + } + + val Zero = DEF() --> (∅.typedWith(𝔹)(empty_in_B), 𝔹) + + val One = { + val One = DEF() --> singleton(∅) + val One_in_B = Theorem( (One :: 𝔹) ) { + have(thesis) by Substitution.ApplyRules(One.shortDefinition)(sing_empty_in_B) + } + One.typedWith(𝔹)(One_in_B) + } + + val zero_in_B = Theorem( (Zero :: 𝔹) ) { + have( Zero :: 𝔹) by TypeChecker.prove + } + + +} + +object TypeSystem { + + import TypeLib.{ + |=>, app, f, x, A, B, funcspaceAxiom, any, definition, given + } + + + + type Class = Term | (Term**1 |-> Formula) + + type SmallClass = Term + + case class FunctionalClass(in: Seq[Class], args: Seq[Variable], out: Class, arity: Int) { + def formula[N <: Arity](f: (Term**N |-> Term)): Formula = + val inner = (args.zip(in.toSeq).map((term, typ) => (term is typ).asFormula).reduceLeft((a, b) => a /\ b)) ==> (f.applySeq(args) is out) + args.foldRight(inner)((v, form) => forall(v, form)) + + override def toString(): String = in.map(_.toStringSeparated()).mkString("(", ", ", ")") + " |> " + out.toStringSeparated() + } + object FunctionalClass { + def apply(in: Seq[Class], out: Class, arity: Int): FunctionalClass = FunctionalClass(in, Seq.empty, out, arity) + } + + extension [N <: Arity](f: (Term**N |-> Term)) { + def is (typ: FunctionalClass): FunctionalTypeAssignment[N] & Formula = FunctionalTypeAssignment[N](f, typ).asInstanceOf + } + + extension[A <: Class] (c: Class) { + //(1 to arity).map(i => freshVariable(in.toSeq :+ out, "%x" + i)) + def |>(out: Class): FunctionalClass = FunctionalClass(Seq(c),Seq(freshVariable(Seq(out), "$x")), out, 1) + def |>(out: FunctionalClass): FunctionalClass = + val newVar = freshVariable(out.in.toSeq :+ out.out, "$x") + FunctionalClass(c +: out.in, newVar +: out.args, out.out, out.arity+1) + def toStringSeparated(): String = c match + case t: Term => t.toStringSeparated() + case f: (Term**1 |-> Formula) @unchecked => f.toString() + } + + extension [A <: Class](t: Term) { + def is(clas:A): Formula with TypeAssignment[A] = TypeAssignment(t, clas).asInstanceOf[Formula with TypeAssignment[A]] + def ::(clas:A): Formula with TypeAssignment[A] = TypeAssignment(t, clas).asInstanceOf[Formula with TypeAssignment[A]] + def @@(t2: Term): AppliedFunction = AppliedFunction(t, t2) + def *(t2: Term): AppliedFunction = AppliedFunction(t, t2) + } + + + object * {def unapply(t: Term): Option[(Term, Term)] = t match { + case AppliedFunction(f, a) => Some((f, a)) + case app(f, a) => Some((f, a)) + case _ => None + }} + + + /** + * A type assumption is a pair of a variable and a type. + * It is also a formula, equal to the type applied to the variable. + */ + sealed trait TypeAssignment[A <: Class]{ + this: Formula => + val t: Term + val typ: A + val asFormula: Formula = this + + override def toString() = + t.toStringSeparated() + "::" + typ.toStringSeparated() + override def toStringSeparated(): String = "(" + toString() + ")" + } + object TypeAssignment { + + /** + * A type assumption is a pair of a variable and a type. + * It is also a formula, equal to the type applied to the variable. + */ + def apply[A <: Class](t: Term, typ:A): TypeAssignment[A] = + val form = typ match + case f: Term => in(t, f) + case f : (Term**1 |-> Formula) @unchecked => + ((f: Term**1 |-> Formula)(t): Formula) + form match + case f: VariableFormula => + throw new IllegalArgumentException("Class formula cannot be a variable formula, as we require a type to have no free variable.") + case f: ConstantFormula => new TypeAssignmentConstant(t, typ, f) + case f: AppliedPredicate => new TypeAssignmentPredicate(t, typ, f) + case f: AppliedConnector => new TypeAssignmentConnector(t, typ, f) + case f: BinderFormula => new TypeAssignmentBinder(t, typ, f) + + def unapply(ta: Formula): Option[(Term, Class)] = ta match + case ta: TypeAssignment[?] => Some((ta.t, ta.typ)) + case in(x, set) => Some((x, set)) + case AppliedPredicate(label, args) if label.arity == 1 => Some((args.head, label.asInstanceOf[Term**1 |-> Formula])) + case _ => None + + } + val is = TypeAssignment + + given [A <: Class]: Conversion[TypeAssignment[A], Formula] = _.asInstanceOf[Formula] + /* + given [A <: Class]: Conversion[TypeAssignment[A], Sequent] = ta => Sequent(Set.empty, Set(ta)) + given [A <: Class]: FormulaSetConverter[TypeAssignment[A]] with { + override def apply(f: TypeAssignment[A]): Set[Formula] = Set(f.asInstanceOf[Formula]) + } +*/ + + + private class TypeAssignmentConstant[A <: Class](val t: Term, val typ:A, formula: ConstantFormula) extends ConstantFormula(formula.id) with TypeAssignment[A] + private class TypeAssignmentPredicate[A <: Class](val t: Term, val typ:A, formula: AppliedPredicate) extends AppliedPredicate(formula.label, formula.args) with TypeAssignment[A] { + + override def substituteUnsafe(map: Map[FOL.SchematicLabel[?], FOL.LisaObject[?]]): FOL.Formula = + if map.keySet.exists(_ == typ) then super.substituteUnsafe(map) + else + val newArgs = args.map(_.substituteUnsafe(map)) + if newArgs == args then this else + val newTyp = (typ: LisaObject[?]).substituteUnsafe(map).asInstanceOf[A] + + TypeAssignmentPredicate(t.substituteUnsafe(map), newTyp, formula.copy(args = newArgs)) + } + private class TypeAssignmentConnector[A <: Class](val t: Term, val typ:A, formula: AppliedConnector) extends AppliedConnector(formula.label, formula.args) with TypeAssignment[A] + private class TypeAssignmentBinder[A <: Class](val t: Term, val typ:A, formula: BinderFormula) extends BinderFormula(formula.f, formula.bound, formula.body) with TypeAssignment[A] + + + type TypingContext = Iterable[TypeAssignment[?]] + + + sealed trait FunctionalTypeAssignment[N <: Arity]{ + this: Formula => + val functional: Term**N |-> Term + val typ: FunctionalClass + val asFormula: Formula = this + + + override def toString() = functional.toString() + " : " + typ.toString() + override def toStringSeparated(): String = "(" + toString + ")" + + } + object FunctionalTypeAssignment { + def apply[N <: Arity](functional: Term**N |-> Term, typ: FunctionalClass): FunctionalTypeAssignment[N] = + val form = typ.formula(functional) + form match + case fo: VariableFormula => + throw new IllegalArgumentException("Class formula cannot be a variable formula, as we require a type to have no free variable.") + case fo: ConstantFormula => new FunctionalTypeAssignmentConstant(functional, typ, fo) + case fo: AppliedPredicate => new FunctionalTypeAssignmentPredicate(functional, typ, fo) + case fo: AppliedConnector => new FunctionalTypeAssignmentConnector(functional, typ, fo) + case fo: BinderFormula => new FunctionalTypeAssignmentBinder(functional, typ, fo) + + def unapply[N <: Arity](f: Formula): Option[((Term ** N) |-> Term, FunctionalClass)] = + f match + case ta: FunctionalTypeAssignment[N] => Some((ta.functional, ta.typ)) + case _ => None + } + private class FunctionalTypeAssignmentConstant[N <: Arity](val functional: Term**N |-> Term, val typ: FunctionalClass, formula: ConstantFormula) extends ConstantFormula(formula.id) with FunctionalTypeAssignment[N] + private class FunctionalTypeAssignmentPredicate[N <: Arity](val functional: Term**N |-> Term, val typ: FunctionalClass, formula: AppliedPredicate) extends AppliedPredicate(formula.label, formula.args) with FunctionalTypeAssignment[N] + private class FunctionalTypeAssignmentConnector[N <: Arity](val functional: Term**N |-> Term, val typ: FunctionalClass, formula: AppliedConnector) extends AppliedConnector(formula.label, formula.args) with FunctionalTypeAssignment[N] + private class FunctionalTypeAssignmentBinder[N <: Arity](val functional: Term**N |-> Term, val typ: FunctionalClass, formula: BinderFormula) extends BinderFormula(formula.f, formula.bound, formula.body) with FunctionalTypeAssignment[N] + + + + + + class TypedConstant[A <: Class] + (id: Identifier, val typ: A, val justif: JUSTIFICATION) extends Constant(id) with LisaObject[TypedConstant[A]] { + val formula = TypeAssignment(this, typ) + assert(justif.statement.left.isEmpty && (justif.statement.right.head == formula)) + + override def substituteUnsafe(map: Map[lisa.fol.FOL.SchematicLabel[?], lisa.fol.FOL.LisaObject[?]]): TypedConstant[A] = this + } + + // Function Labels + + + + + + class TypedConstantFunctional[N <: Arity]( + id : Identifier, + arity: N, + val typ: FunctionalClass, + val justif: JUSTIFICATION + ) extends ConstantFunctionLabel[N](id, arity) with LisaObject[TypedConstantFunctional[N]] { + + override def substituteUnsafe(map: Map[lisa.fol.FOL.SchematicLabel[?], lisa.fol.FOL.LisaObject[?]]): TypedConstantFunctional[N] = this + } + + + + + + class AppliedFunction(val func: Term, val arg: Term) extends AppliedFunctional(app, Seq(func, arg)) with LisaObject[AppliedFunction] { + + override def substituteUnsafe(map: Map[lisa.fol.FOL.SchematicLabel[?], lisa.fol.FOL.LisaObject[?]]): AppliedFunction = AppliedFunction(func.substituteUnsafe(map), arg.substituteUnsafe(map)) + + override def toString(): String = + func match + case AppliedFunction(af @ AppliedFunctional(label, args), t1) if label.id.name == "=:=" => + s"(${t1.toStringSeparated()} =:=_${args.head.toStringSeparated()} ${arg.toStringSeparated()})" + case _ => + func.toStringSeparated() + "*(" + arg.toStringSeparated() + ")" + override def toStringSeparated(): String = toString() + } + object AppliedFunction { + def unapply(af: AppliedFunctional): Option[(Term, Term)] = af match + case AppliedFunctional(label, Seq(func, arg)) if label == app => Some((func, arg)) + case _ => None + } + + + + /////////////////////// + ///// Definitions ///// + /////////////////////// + + + class TypedSimpleConstantDefinition[A <: Class](using om: OutputManager)(fullName: String, line: Int, file: String)( + val expression: Term, + out: Variable, + j: JUSTIFICATION, + val typ:A + ) extends SimpleFunctionDefinition[0](fullName, line, file)(lambda[Term, Term](Seq[Variable](), expression).asInstanceOf, out, j) { + val typingName = "typing_" + fullName + val typingJudgement = THM( label :: typ, typingName, line, file, InternalStatement)({ + have(expression :: typ) by TypeChecker.prove + thenHave(thesis) by lisa.automation.Substitution.ApplyRules(getShortDefinition(label).get) + }) + val typedLabel: TypedConstant[A] = TypedConstant(label.id, typ, typingJudgement) + + + } + object TypedSimpleConstantDefinition { + def apply[A <: Class](using om: OutputManager)(fullName: String, line: Int, file: String)(expression: Term, typ:A): TypedSimpleConstantDefinition[A] = { + val intName = "definition_" + fullName + val out = Variable(freshId(expression.allSchematicLabels.map(_.id), "y")) + val defThm = THM(ExistsOne(out, out === expression), intName, line, file, InternalStatement)({ + have(lisa.prooflib.SimpleDeducedSteps.simpleFunctionDefinition(lambda(Seq[Variable](), expression), out)) + }) + new TypedSimpleConstantDefinition(fullName, line, file)(expression, out, defThm, typ) + } + } + extension (d: definitionWithVars[0]) { + inline infix def -->[A<:Class]( + using om: OutputManager, name: sourcecode.FullName, line: sourcecode.Line, file: sourcecode.File)(term:Term, typ: A): TypedConstant[A] = + TypedSimpleConstantDefinition[A](name.value, line.value, file.value)(term, typ).typedLabel + } + + extension (c: Constant) { + def typedWith[A <: Class](typ:A)(justif: JUSTIFICATION) : TypedConstant[A] = + if justif.statement.right.size != 1 || justif.statement.left.size != 0 || !K.isSame((c is typ).asFormula.underlying, justif.statement.right.head.underlying) then + throw new IllegalArgumentException(s"A proof of typing of $c must be of the form ${c :: typ}, but the given justification shows ${justif.statement}.") + else TypedConstant(c.id, typ, justif) + } + + + + + + + + + ///////////////////////// + ///// Type Checking ///// + ///////////////////////// + + object TypeChecker extends ProofTactic { + private val x = variable + + class TypingException(val msg: String) extends Exception(msg) + + def prove(using proof: SetTheoryLibrary.Proof)(bot:lisa.fol.FOL.Sequent): proof.ProofTacticJudgement = + val context = bot.left + var success: proof.ProofTacticJudgement = null + var typingError: proof.ProofTacticJudgement = null + bot.right.find(goal => + goal match + case (term is typ) => + val ptj = typecheck(using SetTheoryLibrary)(context.toSeq, term, Some(typ)) + if ptj.isValid then + success = ptj + true + else + typingError = ptj + false + case _ => false + ) + if success != null then success else if typingError != null then typingError else proof.InvalidProofTactic("The right hand side of the goal must be a typing judgement") + + private def fullFlat(context: Seq[Formula]): Seq[Formula] = context.flatMap{ + case AppliedConnector(And, cunj) => fullFlat(cunj) + case f => Seq(f) + } + + def typecheck(using lib: SetTheoryLibrary.type, proof: lib.Proof)(context: Seq[Formula], term:Term, typ: Option[Class]): proof.ProofTacticJudgement = + val typingAssumptions: Map[Term, Seq[Class]] = fullFlat(context).collect{ + case TypeAssignment(term, typ) => (term, typ) + }.groupBy(_._1).map((t, l) => (t, l.map(_._2))) + + val functionalTypingAssumptions: Map[(? |-> Term), Seq[FunctionalClass]] = context.collect{ + case FunctionalTypeAssignment(func, typ) => (func, typ) + }.groupBy(_._1).map((func, l) => (func, l.map(_._2))) + + TacticSubproof { + context.foreach(assume(_)) + try { + + def innerTypecheck(context2: Map[Term, Seq[Class]], term:Term, typ:Option[Class]): Class= { + val possibleTypes = typingAssumptions.getOrElse(term, Nil) + if typ == Some(any) then + have(term is any) by Restate.from(TypeLib.any.definition of (x := term)) + any + else if typ.isEmpty && possibleTypes.size >=1 then + have(term is possibleTypes.head) by Restate + possibleTypes.head + else if (typ.nonEmpty && possibleTypes.contains(typ.get)) then + have(term is typ.get) by Restate + typ.get + else term match + case tc: TypedConstant[?] => + if typ.isEmpty then + have(tc is tc.typ) by Restate.from(tc.justif) + tc.typ + else if K.isSame((tc is typ.get).asFormula.underlying, (tc is tc.typ).asFormula.underlying) then + have(tc is typ.get) by Restate.from(tc.justif) + typ.get + else throw TypingException("Constant " + tc + " expected to be of type " + typ + " but has type " + tc.typ + ".") + + case AppliedFunction(func, arg) => + val funcType = innerTypecheck(context2, func, None) + val funcProof = lastStep + val argType = innerTypecheck(context2, arg, None) + val argProof = lastStep + funcType match + case inType |=> outType => typ match + case None => + if K.isSame((arg is inType).asFormula.underlying, (arg is argType).asFormula.underlying) then + have(term is outType) by Tautology.from( + funcspaceAxiom of (f := func, x := arg, A:= inType, B:= outType), + funcProof, + argProof + ) + outType + else throw + TypingException("Function " + func + " found to have type " + funcType + ", but argument " + arg + " has type " + argType + " instead of expected " + inType + ".") + case Some(typ) if K.isSame((term is typ).asFormula.underlying, (term is outType).asFormula.underlying) => + if K.isSame((arg is inType).asFormula.underlying, (arg is argType).asFormula.underlying) then + have(term is outType) by Tautology.from( + funcspaceAxiom of (f := func, x := arg, A:= inType, B:= outType), + funcProof, + argProof + ) + typ + else + throw TypingException("Function " + func + " found to have type " + funcType + ", but argument " + arg + " has type " + argType + " instead of expected " + inType + ".") + + case _ => + throw TypingException("Function " + func + " expected to have function type ? |=> " + typ + ", but has type " + funcType + ". ") + case _ => + throw TypingException("Function " + func + " expected to have function type ? |=> " + typ + ", but has type " + funcType + ". Note that terms having multiple different types is only partialy supported.") + + case AppliedFunctional(label, args) => + val (argTypes, argTypesProofs) = args.map(arg => + try (innerTypecheck(context2, arg, None), lastStep) + catch + case e: TypingException => (any, any.definition of (x := arg)) //if no type could be constructed the normal way, we give it "any" + ).unzip + val labelTypes = label match + case label: TypedConstantFunctional[?] => + (label.typ, () => label.justif) +: + functionalTypingAssumptions.getOrElse(label, Nil).map(fc => (fc, () => (have( fc.formula(label.asInstanceOf) ) by Restate) )) + + case _ => functionalTypingAssumptions.getOrElse(label, Nil).map(fc => (fc, () => have( fc.formula(label.asInstanceOf) ) by Restate )) + functionalTypingAssumptions.get(label) + if labelTypes.isEmpty then + throw TypingException("Function " + label + " expected to have type (" + argTypes.mkString(", ") + ") |=> ? but is untyped.") + else + typ match + case None => + labelTypes.find((labelType, step) => + labelType.arity == args.size && + (args zip argTypes).zip(labelType.in.toSeq).forall((argAndTypes, inType) => + K.isSame((argAndTypes._1 is inType).asFormula.underlying, (argAndTypes._1 is argAndTypes._2).asFormula.underlying) // + ) + ) match + case None => + throw TypingException("Function " + label + " expected to have type (" + argTypes.mkString(", ") + ") |=> ? but is assigned " + labelTypes.mkString(" & ") + ". Note that terms having multiple different types is only partialy supported.") + case Some(labelType, step) => + val out: Class = labelType.out + + val in: Seq[Class] = labelType.in.toSeq + //val labelProp = labelType.formula(label.asInstanceOf) + val labelPropStatement = step() + val labInst = labelPropStatement.of(args: _*) + val subst = (labelType.args zip args).map((v, a) => (v := a)) + val newOut: Class = out match { + case t: Term => t.substitute(subst: _*) + case f: (Term**1 |-> Formula) @unchecked => f.substitute(subst: _*) + } + have(term is newOut) by Tautology.from( + (argTypesProofs :+ labInst ) : _* + ) + newOut + case Some(typValue) => + labelTypes.find((labelType, step) => + labelType.arity == args.size && + (args zip argTypes).zip(labelType.in.toSeq).forall((argAndTypes, inType) => + K.isSame((argAndTypes._1 is inType).asFormula.underlying, (argAndTypes._1 is argAndTypes._2).asFormula.underlying) + ) && { + val subst = (labelType.args zip args).map((v, a) => (v := a)) + val newOut: Class = labelType.out match { + case t: Term => t.substitute(subst: _*) + case f: (Term**1 |-> Formula) @unchecked => f.substitute(subst: _*) + } + K.isSame((term is newOut).asFormula.underlying, (term is typValue).asFormula.underlying) + + } + ) match + case None => + throw TypingException("Function " + label + " expected to have type (" + argTypes.mkString(", ") + ") |=> " + typValue + " but is assigned " + labelTypes.mkString(" & ") + ". Note that terms having multiple different types is only partialy supported.") + case Some(labelType, step) => + val out: Class = labelType.out + val in: Seq[Class] = labelType.in.toSeq + //val labelProp = labelType.formula(label.asInstanceOf) + val labelPropStatement = step() + have(term is typValue) by Tautology.from( + (argTypesProofs :+ labelPropStatement.of(args: _*) ) : _* + ) + typValue + + case v: Variable => + if possibleTypes.isEmpty then + throw TypingException("Variable " + v + " expected to be of type " + typ + " but is untyped.") + else throw TypingException("Variable " + v + " expected to be of type " + typ + " but is assigned " + possibleTypes.mkString(" & ") + ".") + + case c: Constant => + if possibleTypes.isEmpty then + throw TypingException("Constant " + c + " expected to be of type " + typ + " but is untyped.") + else throw TypingException("Constant " + c + " expected to be of type " + typ + " but is assigned " + possibleTypes.mkString(" & ") + ".") + + case _: AppliedFunctional => + throw Exception("Why is this not handled by the previous case? Scala reports an incomplete match") + + } + innerTypecheck(typingAssumptions, term, typ) + } + catch { + case e: TypingException => + return proof.InvalidProofTactic(e.msg) + } + } + + + } + + + + + + + + + + +} diff --git a/lisa-utils/src/main/scala/lisa/fol/Common.scala b/lisa-utils/src/main/scala/lisa/fol/Common.scala index cb1ca1eeb..c125f7580 100644 --- a/lisa-utils/src/main/scala/lisa/fol/Common.scala +++ b/lisa-utils/src/main/scala/lisa/fol/Common.scala @@ -135,6 +135,8 @@ trait Common { * Renames the symbol with an identifier that is fresh for the given list. */ def freshRename(taken: Iterable[Identifier]): Label[A] + + } /** @@ -276,7 +278,7 @@ trait Common { sealed trait FunctionLabel[N <: Arity] extends TermLabel[(Term ** N) |-> Term] with ((Term ** N) |-> Term) { val underlyingLabel: K.TermLabel def substituteUnsafe(map: Map[SchematicLabel[_], LisaObject[_]]): (Term ** N) |-> Term - def applyUnsafe(args: (Term ** N)): Term = AppliedFunction(this, args.toSeq) + def applyUnsafe(args: (Term ** N)): Term = AppliedFunctional(this, args.toSeq) override def rename(newid: Identifier): FunctionLabel[N] def freshRename(taken: Iterable[Identifier]): FunctionLabel[N] } @@ -285,7 +287,7 @@ trait Common { * A Variable, corresponding to [[K.VariableLabel]], is a schematic symbol for terms. * It counts both as the label and as the term itself. */ - case class Variable(id: Identifier) extends SchematicTermLabel[Term] with Term with Absolute { + class Variable(val id: Identifier) extends SchematicTermLabel[Term] with Term with Absolute { val arity: 0 = 0 val label: Variable = this val args: Seq[Nothing] = Seq.empty @@ -308,13 +310,38 @@ trait Common { def freshRename(taken: Iterable[Identifier]): Variable = rename(K.freshId(taken, id)) override def toString(): String = id def mkString(args: Seq[Term]): String = if (args.size == 0) toString() else toString() + "(" + "illegal_arguments: " + args.mkString(", ") + ")" + + def canEqual(that: Any): Boolean = + that.isInstanceOf[Variable] + + //Intentionally avoiding the call to super.equals because no ancestor has overridden equals (see note 7 below) + override def equals(that: Any): Boolean = + that match { + case other: Variable => + ( (this eq other) //optional, but highly recommended sans very specific knowledge about this exact class implementation + || ( other.canEqual(this) //optional only if this class is marked final + && (hashCode == other.hashCode) //optional, exceptionally execution efficient if hashCode is cached, at an obvious space inefficiency tradeoff + && ( (id == other.id) + ) + ) + ) + case _ => + false + } + + //Intentionally avoiding the call to super.hashCode because no ancestor has overridden hashCode (see note 7 below) + override def hashCode(): Int = + id.## + } + object Variable { + def unapply(variable: Variable): Option[Identifier] = Some(variable.id) } /** * A Constant, corresponding to [[K.ConstantLabel]], is a label for terms. * It counts both as the label and as the term itself. */ - case class Constant(id: Identifier) extends Term with Absolute with ConstantTermLabel[Constant] with LisaObject[Constant] { + class Constant(val id: Identifier) extends Term with Absolute with ConstantTermLabel[Constant] with LisaObject[Constant] { val arity: 0 = 0 val label: Constant = this val args: Seq[Nothing] = Seq.empty @@ -328,17 +355,42 @@ trait Common { def freshRename(taken: Iterable[Identifier]): Constant = rename(K.freshId(taken, id)) override def toString(): String = id def mkString(args: Seq[Term]): String = if (args.size == 0) toString() else toString() + "(" + "illegal_arguments: " + args.mkString(", ") + ")" + + def canEqual(that: Any): Boolean = + that.isInstanceOf[Constant] + + //Intentionally avoiding the call to super.equals because no ancestor has overridden equals (see note 7 below) + override def equals(that: Any): Boolean = + that match { + case other: Constant => + ( (this eq other) //optional, but highly recommended sans very specific knowledge about this exact class implementation + || ( other.canEqual(this) //optional only if this class is marked final + && (hashCode == other.hashCode) //optional, exceptionally execution efficient if hashCode is cached, at an obvious space inefficiency tradeoff + && ( (id == other.id) + ) + ) + ) + case _ => + false + } + + //Intentionally avoiding the call to super.hashCode because no ancestor has overridden hashCode (see note 7 below) + override def hashCode(): Int = + id.## + } + object Constant { + def unapply(constant: Constant): Option[Identifier] = Some(constant.id) } /** * A schematic functional label (corresponding to [[K.SchematicFunctionLabel]]) is a functional label and also a schematic label. * It can be substituted by any expression of type (Term ** N) |-> Term */ - case class SchematicFunctionLabel[N <: Arity](val id: Identifier, val arity: N) extends SchematicTermLabel[(Term ** N) |-> Term] with FunctionLabel[N] { + class SchematicFunctionLabel[N <: Arity](val id: Identifier, val arity: N) extends SchematicTermLabel[(Term ** N) |-> Term] with FunctionLabel[N] { val underlyingLabel: K.SchematicTermLabel = K.SchematicFunctionLabel(id, arity) - def unapplySeq(t: AppliedFunction): Seq[Term] = t match { - case AppliedFunction(label, args) if (label == this) => args + def unapplySeq(t: AppliedFunctional): Seq[Term] = t match { + case AppliedFunctional(label, args) if (label == this) => args case _ => Seq.empty } @nowarn("msg=the type test for.*cannot be checked at runtime because its type arguments") @@ -359,16 +411,44 @@ trait Common { override def toString(): String = id def mkString(args: Seq[Term]): String = toString() + "(" + args.mkString(", ") + ")" override def mkStringSeparated(args: Seq[Term]): String = mkString(args) + + def canEqual(that: Any): Boolean = + that.isInstanceOf[SchematicFunctionLabel[?]] + + //Intentionally avoiding the call to super.equals because no ancestor has overridden equals (see note 7 below) + override def equals(that: Any): Boolean = + that match { + case other: SchematicFunctionLabel[N] => + ( (this eq other) //optional, but highly recommended sans very specific knowledge about this exact class implementation + || ( other.canEqual(this) //optional only if this class is marked final + && (hashCode == other.hashCode) //optional, exceptionally execution efficient if hashCode is cached, at an obvious space inefficiency tradeoff + && ( (id == other.id) + && (arity == other.arity) + ) + ) + ) + case _ => + false + } + + //Intentionally avoiding the call to super.hashCode because no ancestor has overridden hashCode (see note 7 below) + override def hashCode(): Int = + 31 * ( + id.## + ) + arity.## + } + object SchematicFunctionLabel { + def unapply[N <: Arity](sfl: SchematicFunctionLabel[N]): Option[(Identifier, N)] = Some((sfl.id, sfl.arity)) } /** * A constant functional label of arity N. */ - case class ConstantFunctionLabel[N <: Arity](id: Identifier, arity: N) extends ConstantTermLabel[((Term ** N) |-> Term)] with FunctionLabel[N] { + class ConstantFunctionLabel[N <: Arity](val id: Identifier, val arity: N) extends ConstantTermLabel[((Term ** N) |-> Term)] with FunctionLabel[N] { val underlyingLabel: K.ConstantFunctionLabel = K.ConstantFunctionLabel(id, arity) private var infix: Boolean = false - def unapplySeq(t: AppliedFunction): Seq[Term] = t match { - case AppliedFunction(label, args) if (label == this) => args + def unapplySeq(t: AppliedFunctional): Seq[Term] = t match { + case AppliedFunctional(label, args) if (label == this) => args case _ => Seq.empty } def substituteUnsafe(map: Map[SchematicLabel[_], LisaObject[_]]): ConstantFunctionLabel[N] = this @@ -377,20 +457,46 @@ trait Common { def rename(newid: Identifier): ConstantFunctionLabel[N] = ConstantFunctionLabel(newid, arity) def freshRename(taken: Iterable[Identifier]): ConstantFunctionLabel[N] = rename(K.freshId(taken, id)) override def toString(): String = id - def mkString(args: Seq[Term]): String = if (infix) (args(0).toStringSeparated() + " " + toString() + " " + args(1).toStringSeparated()) else toString() + "(" + args.mkString(", ") + ")" + def mkString(args: Seq[Term]): String = if (infix & args.size == 2) (args(0).toStringSeparated() + " " + toString() + " " + args(1).toStringSeparated()) else toString() + "(" + args.mkString(", ") + ")" override def mkStringSeparated(args: Seq[Term]): String = if (infix) "(" + mkString(args) + ")" else mkString(args) + + def canEqual(that: Any): Boolean = + that.isInstanceOf[SchematicFunctionLabel[?]] + + //Intentionally avoiding the call to super.equals because no ancestor has overridden equals (see note 7 below) + override def equals(that: Any): Boolean = + that match { + case other: ConstantFunctionLabel[N] => + ( (this eq other) //optional, but highly recommended sans very specific knowledge about this exact class implementation + || ( other.canEqual(this) //optional only if this class is marked final + && (hashCode == other.hashCode) //optional, exceptionally execution efficient if hashCode is cached, at an obvious space inefficiency tradeoff + && ( (id == other.id) + && (arity == other.arity) + ) + ) + ) + case _ => + false + } + + //Intentionally avoiding the call to super.hashCode because no ancestor has overridden hashCode (see note 7 below) + override def hashCode(): Int = + 31 * ( + id.## + ) + arity.## } object ConstantFunctionLabel { def infix[N <: Arity](id: Identifier, arity: N): ConstantFunctionLabel[N] = val x = ConstantFunctionLabel[N](id, arity) x.infix = true x + def unapply[N <: Arity](cfl: ConstantFunctionLabel[N]): Option[(Identifier, N)] = Some((cfl.id, cfl.arity)) } /** * A term made from a functional label of arity N and N arguments */ - case class AppliedFunction(label: FunctionLabel[?], args: Seq[Term]) extends Term with Absolute { + class AppliedFunctional(val label: FunctionLabel[?], val args: Seq[Term]) extends Term with Absolute { override val underlying = K.Term(label.underlyingLabel, args.map(_.underlying)) def substituteUnsafe(map: Map[SchematicLabel[_], LisaObject[_]]): Term = label.substituteUnsafe(map).applyUnsafe(args.map[Term]((x: Term) => x.substituteUnsafe(map))) @@ -399,6 +505,34 @@ trait Common { def allSchematicLabels: Set[SchematicLabel[?]] = label.allSchematicLabels ++ args.flatMap(_.allSchematicLabels) override def toString: String = label.mkString(args) override def toStringSeparated(): String = label.mkStringSeparated(args) + + def canEqual(that: Any): Boolean = + that.isInstanceOf[AppliedFunctional] + + //Intentionally avoiding the call to super.equals because no ancestor has overridden equals (see note 7 below) + override def equals(that: Any): Boolean = + that match { + case other: AppliedFunctional => + ( (this eq other) //optional, but highly recommended sans very specific knowledge about this exact class implementation + || ( other.canEqual(this) //optional only if this class is marked final + && (hashCode == other.hashCode) //optional, exceptionally execution efficient if hashCode is cached, at an obvious space inefficiency tradeoff + && ( (label == other.label) + && (args == other.args) + ) + ) + ) + case _ => + false + } + + //Intentionally avoiding the call to super.hashCode because no ancestor has overridden hashCode (see note 7 below) + override def hashCode(): Int = + 31 * ( + label.## + ) + args.## + } + object AppliedFunctional { + def unapply(af: AppliedFunctional): Option[(FunctionLabel[?], Seq[Term])] = Some((af.label, af.args)) } ////////////////////////////////////// @@ -570,8 +704,8 @@ trait Common { */ case class SchematicPredicateLabel[N <: Arity](id: Identifier, arity: N) extends SchematicAtomicLabel[(Term ** N) |-> Formula] with PredicateLabel[N] { val underlyingLabel: K.SchematicPredicateLabel = K.SchematicPredicateLabel(id, arity) - def unapplySeq(t: AppliedFunction): Seq[Term] = t match { - case AppliedFunction(label, args) if (label == this) => args + def unapplySeq(t: AppliedFunctional): Seq[Term] = t match { + case AppliedFunctional(label, args) if (label == this) => args case _ => Seq.empty } @nowarn("msg=the type test for.*cannot be checked at runtime because its type arguments") @@ -808,4 +942,5 @@ trait Common { def apply(s1: S, s2: S, s3: S, s4: S, s5: S): T = t.applyUnsafe(Seq(s1, s2, s3, s4, s5)) } -} + +} \ No newline at end of file diff --git a/lisa-utils/src/main/scala/lisa/fol/FOLHelpers.scala b/lisa-utils/src/main/scala/lisa/fol/FOLHelpers.scala index 576a279c3..9feb1faec 100644 --- a/lisa-utils/src/main/scala/lisa/fol/FOLHelpers.scala +++ b/lisa-utils/src/main/scala/lisa/fol/FOLHelpers.scala @@ -130,4 +130,7 @@ object FOLHelpers { def asFrontLambda(l: K.LambdaTermFormula): LambdaExpression[Term, Formula, ?] = LambdaExpression(l.vars.map(asFrontLabel), asFront(l.body), l.vars.size) def asFrontLambda(l: K.LambdaFormulaFormula): LambdaExpression[Formula, Formula, ?] = LambdaExpression(l.vars.map(asFrontLabel), asFront(l.body), l.vars.size) + + def freshVariable[A <: LisaObject[A]](obj: A, name: Identifier): Variable = Variable(freshId(obj.allSchematicLabels.map(_.id), name)) + def freshVariable[A <: LisaObject[A]](objs: Iterable[A], name: Identifier): Variable = Variable(freshId(objs.flatMap(_.allSchematicLabels).map(_.id), name)) } diff --git a/lisa-utils/src/main/scala/lisa/fol/Lambdas.scala b/lisa-utils/src/main/scala/lisa/fol/Lambdas.scala index 2bc2a9840..bfc152375 100644 --- a/lisa-utils/src/main/scala/lisa/fol/Lambdas.scala +++ b/lisa-utils/src/main/scala/lisa/fol/Lambdas.scala @@ -36,7 +36,7 @@ trait Lambdas extends Common { */ def substituteUnsafe(map: Map[SchematicLabel[_], LisaObject[_]]): LambdaExpression[T, R, N] = { val newSubst = map -- seqBounds - val conflict = map.values.flatMap(_.freeSchematicLabels).toSet.intersect(bounds.asInstanceOf) + val conflict = map.values.flatMap(_.freeSchematicLabels).toSet.intersect(bounds.toSet.asInstanceOf) if (conflict.nonEmpty) { val taken = (map.values.flatMap(_.allSchematicLabels).map(_.id) ++ map.keys.map(_.id)).toList val newBounds = seqBounds.scanLeft[List[Identifier]](taken)((list, v: SchematicLabel[T]) => freshId(list, v.id) :: list).map(_.head).zip(seqBounds).map(v => v._2.rename(v._1)) diff --git a/lisa-utils/src/main/scala/lisa/fol/Sequents.scala b/lisa-utils/src/main/scala/lisa/fol/Sequents.scala index 1f13bf849..792c75aa1 100644 --- a/lisa-utils/src/main/scala/lisa/fol/Sequents.scala +++ b/lisa-utils/src/main/scala/lisa/fol/Sequents.scala @@ -17,8 +17,8 @@ trait Sequents extends Common with lisa.fol.Lambdas with Predef { case class Sequent(left: Set[Formula], right: Set[Formula]) extends LisaObject[Sequent] with Absolute { def underlying: lisa.kernel.proof.SequentCalculus.Sequent = K.Sequent(left.map(_.underlying), right.map(_.underlying)) - def allSchematicLabels: Set[SchematicLabel[?]] = left.flatMap(_.allSchematicLabels) - def freeSchematicLabels: Set[SchematicLabel[?]] = left.flatMap(_.freeSchematicLabels) + def allSchematicLabels: Set[SchematicLabel[?]] = left.flatMap(_.allSchematicLabels) ++ right.flatMap(_.allSchematicLabels) + def freeSchematicLabels: Set[SchematicLabel[?]] = left.flatMap(_.freeSchematicLabels) ++ right.flatMap(_.freeSchematicLabels) def substituteUnsafe(map: Map[SchematicLabel[?], ? <: LisaObject[?]]): Sequent = Sequent(left.map(_.substituteUnsafe(map)), right.map(_.substituteUnsafe(map))) /*Ok for now but what when we have more*/ @@ -89,7 +89,7 @@ trait Sequents extends Common with lisa.fol.Lambdas with Predef { case (s, p) => (s, Seq(s0, s1, s2) ++ p) } - case _ => throw new IllegalArgumentException("Right side of sequent must be a single universaly quantified formula") + case _ => throw new IllegalArgumentException("Right side of sequent must be a single universally quantified formula") } } @@ -215,7 +215,7 @@ trait Sequents extends Common with lisa.fol.Lambdas with Predef { * @tparam S The type of elements in that set * @tparam T The type to convert from */ - protected trait FormulaSetConverter[T] { + trait FormulaSetConverter[T] { def apply(t: T): Set[Formula] } diff --git a/lisa-utils/src/main/scala/lisa/prooflib/BasicMain.scala b/lisa-utils/src/main/scala/lisa/prooflib/BasicMain.scala index f7bcd5f8f..34f8878fd 100644 --- a/lisa-utils/src/main/scala/lisa/prooflib/BasicMain.scala +++ b/lisa-utils/src/main/scala/lisa/prooflib/BasicMain.scala @@ -15,6 +15,7 @@ trait BasicMain { } val stringWriter: java.io.StringWriter = new java.io.StringWriter() } + export om.output /** * This specific implementation make sure that what is "shown" in theory files is only printed for the one we run, and not for the whole library. diff --git a/lisa-utils/src/main/scala/lisa/prooflib/BasicStepTactic.scala b/lisa-utils/src/main/scala/lisa/prooflib/BasicStepTactic.scala index 678cd18cd..0501f1b40 100644 --- a/lisa-utils/src/main/scala/lisa/prooflib/BasicStepTactic.scala +++ b/lisa-utils/src/main/scala/lisa/prooflib/BasicStepTactic.scala @@ -446,6 +446,7 @@ object BasicStepTactic { proof.ValidProofTactic(bot, Seq(K.LeftExists(botK, -1, phiK, xK)), Seq(premise)) } + var debug = false def apply(using lib: Library, proof: lib.Proof)(premise: proof.Fact)(bot: F.Sequent): proof.ProofTacticJudgement = { lazy val premiseSequent = proof.getSequent(premise) lazy val pivot = bot.left.diff(premiseSequent.left) @@ -470,14 +471,14 @@ object BasicStepTactic { case Some(F.BinderFormula(F.Exists, x, phi)) => LeftExists.withParameters(phi, x)(premise)(bot) case _ => proof.InvalidProofTactic("Could not infer an existensially quantified pivot from premise and conclusion.") } - } else proof.InvalidProofTactic("Left-hand side of conclusion + φ is not the same as left-hand side of premise + ∃x. φ.") + } else proof.InvalidProofTactic("Ambigous application of LeftExists, multiple pivots corresponding to the unquantified formula found.") else if (pivot.tail.isEmpty) pivot.head match { case F.BinderFormula(F.Exists, x, phi) => LeftExists.withParameters(phi, x)(premise)(bot) case _ => proof.InvalidProofTactic("Could not infer an existentially quantified pivot from premise and conclusion.") } else - proof.InvalidProofTactic("Left-hand side of conclusion + φ is not the same as left-hand side of premise + ∃x. φ.") + proof.InvalidProofTactic("Ambigous application of LeftExists, multiple pivots corresponding to the quantified formula found.") } } @@ -1440,7 +1441,7 @@ object BasicStepTactic { } } - // TODO make specific support for subproofs written inside tactics. + // TODO make specific support for subproofs written inside tactics.kkkkkkk inline def TacticSubproof(using proof: Library#Proof)(inline computeProof: proof.InnerProof ?=> Unit): proof.ProofTacticJudgement = val iProof: proof.InnerProof = new proof.InnerProof(None) diff --git a/lisa-utils/src/main/scala/lisa/prooflib/Library.scala b/lisa-utils/src/main/scala/lisa/prooflib/Library.scala index 8927c4133..fb3a27523 100644 --- a/lisa-utils/src/main/scala/lisa/prooflib/Library.scala +++ b/lisa-utils/src/main/scala/lisa/prooflib/Library.scala @@ -42,6 +42,7 @@ abstract class Library extends lisa.prooflib.WithTheorems with lisa.prooflib.Pro else _draft = Some(file) val knownDefs: scala.collection.mutable.Map[F.ConstantLabel[?], Option[JUSTIFICATION]] = scala.collection.mutable.Map.empty + val shortDefs: scala.collection.mutable.Map[F.ConstantLabel[?], Option[JUSTIFICATION]] = scala.collection.mutable.Map.empty def addSymbol(s: F.ConstantFunctionLabel[?] | F.ConstantPredicateLabel[?] | F.Constant): Unit = { s match { @@ -56,6 +57,10 @@ abstract class Library extends lisa.prooflib.WithTheorems with lisa.prooflib.Pro case None => throw new UserLisaException.UndefinedSymbolException("Unknown symbol", label, this) case Some(value) => value } + def getShortDefinition(label: F.ConstantLabel[?]): Option[JUSTIFICATION] = shortDefs.get(label) match { + case None => throw new UserLisaException.UndefinedSymbolException("Unknown symbol", label, this) + case Some(value) => value + } /** * An alias to create a Theorem diff --git a/lisa-utils/src/main/scala/lisa/prooflib/OutputManager.scala b/lisa-utils/src/main/scala/lisa/prooflib/OutputManager.scala index ce3f84f8b..7cabafea1 100644 --- a/lisa-utils/src/main/scala/lisa/prooflib/OutputManager.scala +++ b/lisa-utils/src/main/scala/lisa/prooflib/OutputManager.scala @@ -16,7 +16,7 @@ abstract class OutputManager { def finishOutput(exception: Exception): Nothing - def lisaThrow(le: LisaException): Nothing = + def lisaThrow(le: LisaException): Nothing = throw le /* le match { case ule: UserLisaException => ule.fixTrace() @@ -31,7 +31,7 @@ abstract class OutputManager { output(e.underlying.repr) finishOutput(e) - } + }*/ def log(e: Exception): Unit = { stringWriter.write("\n[" + Console.RED + "Error" + Console.RESET + "] ") diff --git a/lisa-utils/src/main/scala/lisa/prooflib/ProofsHelpers.scala b/lisa-utils/src/main/scala/lisa/prooflib/ProofsHelpers.scala index e612d0eba..fb39d4551 100644 --- a/lisa-utils/src/main/scala/lisa/prooflib/ProofsHelpers.scala +++ b/lisa-utils/src/main/scala/lisa/prooflib/ProofsHelpers.scala @@ -10,6 +10,7 @@ import lisa.utils.LisaException import lisa.utils.UserLisaException import lisa.utils.parsing.FOLPrinter import lisa.utils.{_, given} +import lisa.kernel.proof.SCProofChecker.checkSCProof import scala.annotation.targetName @@ -20,13 +21,13 @@ trait ProofsHelpers { given Library = library - class HaveSequent private[ProofsHelpers] (bot: Sequent) { - val x: lisa.fol.FOL.Sequent = bot + class HaveSequent /*private[ProofsHelpers]*/ (val bot: Sequent) { + //val x: lisa.fol.FOL.Sequent = bot 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) { - private val bot = HaveSequent.this.bot ++ (F.iterable_to_set(_proof.getAssumptions) |- ()) + val bot = HaveSequent.this.bot ++ (F.iterable_to_set(_proof.getAssumptions) |- ()) inline infix def apply(tactic: Sequent => _proof.ProofTacticJudgement): _proof.ProofStep & _proof.Fact = { tactic(bot).validate(line, file) } @@ -44,7 +45,7 @@ trait ProofsHelpers { } - class AndThenSequent private[ProofsHelpers] (bot: Sequent) { + class AndThenSequent private[ProofsHelpers] (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[By { val _proof: proof.type }] @@ -290,7 +291,17 @@ trait ProofsHelpers { val lambda: LambdaExpression[Term, Term, N], out: F.Variable, j: JUSTIFICATION - ) extends FunctionDefinition[N](fullName, line, file)(lambda.bounds.asInstanceOf, out, out === lambda.body, j) {} + ) extends FunctionDefinition[N](fullName, line, file)(lambda.bounds.asInstanceOf, out, out === lambda.body, j) { + + private val term = label.applySeq(lambda.bounds.asInstanceOf) + private val simpleProp = lambda.body === term + val simplePropName = "simpleDef_" + fullName + val simpleDef = THM(simpleProp, simplePropName, line, file, InternalStatement)({ + have(thesis) by Restate.from(this of term) + }) + shortDefs.update(label, Some(simpleDef)) + + } object SimpleFunctionDefinition { def apply[N <: F.Arity](using om: OutputManager)(fullName: String, line: Int, file: String)(lambda: LambdaExpression[Term, Term, N]): SimpleFunctionDefinition[N] = { @@ -416,4 +427,17 @@ trait ProofsHelpers { } + /** + * Check correctness of the proof, using LISA's logical kernel, to the current point. + */ + def sanityProofCheck(using p: Proof)(message: String): Unit = { + val csc = p.toSCProof + if checkSCProof(csc).isValid then + println("Proof is valid. " + message) + Thread.sleep(100) + else + checkProof(csc) + throw Exception("Proof is not valid: " + message) + } + } diff --git a/lisa-utils/src/main/scala/lisa/prooflib/SimpleDeducedSteps.scala b/lisa-utils/src/main/scala/lisa/prooflib/SimpleDeducedSteps.scala index 8e470c143..eb9b3fc5c 100644 --- a/lisa-utils/src/main/scala/lisa/prooflib/SimpleDeducedSteps.scala +++ b/lisa-utils/src/main/scala/lisa/prooflib/SimpleDeducedSteps.scala @@ -50,22 +50,24 @@ object SimpleDeducedSteps { object Discharge extends ProofTactic { def apply(using lib: Library, proof: lib.Proof)(premises: proof.Fact*)(premise: proof.Fact): proof.ProofTacticJudgement = { - val ss = premises map (e => proof.getSequent(e)) - val seqs = ss map (e => e.underlying) + val ss = premises zip (premises map (e => proof.getSequent(e))) + val seqs = ss.map(_._2) if (!seqs.forall(_.right.size == 1)) return proof.InvalidProofTactic("When discharging this way, the discharged sequent must have only a single formula on the right handside.") - val s = seqs.head - val f = s.right.head - val first = K.Cut((proof.getSequent(premise).underlying removeLeft f) ++ (s removeRight f), -2, -1, f) - - proof.ValidProofTactic( - (proof.getSequent(premise) removeAllLeft (ss.flatMap(_.right).toSet)) ++<< (F.Sequent(ss.flatMap(_.left).toSet, Set())), - seqs.tail.zipWithIndex.scanLeft(first)((prev, next) => { - val f = next._1.right.head - K.Cut((prev.bot removeLeft f) ++ (next._1 removeRight f), -next._2 - 3, next._2, f) - }), - proof.mostRecentStep +: premises - ) + val seqAny = ss.find((_, s) => premise.statement.left.exists(f2 => F.isSame(s.right.head, f2))) + if (seqAny.isEmpty) + Restate.from(premise)(premise.statement) + else + TacticSubproof: ip ?=> + ss.foldLeft(premise: ip.Fact)( + (prem, discharge) => + val seq = discharge._2 + if prem.statement.left.exists(f => F.isSame(f, seq.right.head)) then + val goal = prem.statement - List[K.SCProofStep])] = Nil + def cleanAssumptions: Unit = assumptions = Nil + /** * the theorem that is being proved (paritally, if subproof) by this proof. * @@ -560,7 +562,6 @@ trait WithTheorems { val innerJustification: theory.Theorem = if library._draft.nonEmpty && library._draft.get.value != file then // if the draft option is activated, and the theorem is not in the file where the draft option is given, then we replace the proof by sorry - // println("skip!") theory.theorem(name, goal.underlying, SCProof(SC.Sorry(goal.underlying)), IndexedSeq.empty) match { case K.Judgement.ValidJustification(just) => just diff --git a/lisa-utils/src/main/scala/lisa/utils/KernelHelpers.scala b/lisa-utils/src/main/scala/lisa/utils/KernelHelpers.scala index 122f418d0..70d404521 100644 --- a/lisa-utils/src/main/scala/lisa/utils/KernelHelpers.scala +++ b/lisa-utils/src/main/scala/lisa/utils/KernelHelpers.scala @@ -348,7 +348,7 @@ object KernelHelpers { def theorem(name: String, statement: Sequent, proof: SCProof, justifications: Seq[theory.Justification]): RunningTheoryJudgement[theory.Theorem] = { if (statement == proof.conclusion) theory.makeTheorem(name, statement, proof, justifications) else if (isSameSequent(statement, proof.conclusion)) theory.makeTheorem(name, statement, proof.appended(Restate(statement, proof.length - 1)), justifications) - else InvalidJustification(s"The proof proves ${FOLPrinter.prettySequent(proof.conclusion)} instead of claimed ${FOLPrinter.prettySequent(statement)}", None) + else InvalidJustification(s"The proof proves \n ${FOLPrinter.prettySequent(proof.conclusion)}\ninstead of claimed \n ${FOLPrinter.prettySequent(statement)}", None) } /** diff --git a/lisa-utils/src/main/scala/lisa/utils/memoization/Memoized.scala b/lisa-utils/src/main/scala/lisa/utils/memoization/Memoized.scala new file mode 100644 index 000000000..18525eeea --- /dev/null +++ b/lisa-utils/src/main/scala/lisa/utils/memoization/Memoized.scala @@ -0,0 +1,38 @@ +package lisa.utils.memoization + +case class MemoizationStats(hits: Int, miss: Int, faulted: Int): + def withHit = MemoizationStats(hits + 1, miss, faulted) + def withMiss = MemoizationStats(hits, miss + 1, faulted) + def withFault = MemoizationStats(hits, miss, faulted + 1) + +case object InfiniteRecursionDetectedException extends Exception + +class Memoized[From, To](fun: From => To) extends Function[From, To]: + private val visited = scala.collection.mutable.HashSet.empty[From] + private val memory = scala.collection.mutable.HashMap.empty[From, To] + private var stats = MemoizationStats(0, 0, 0) + + protected def handleFault(): To = + throw InfiniteRecursionDetectedException + + def apply(v: From): To = + val stored = memory.get(v) + val seen = visited.contains(v) + if stored.isEmpty then + // compute + visited.add(v) + if seen then + stats = stats.withFault + handleFault() + else + stats = stats.withMiss + memory.update(v, fun(v)) + else + stats = stats.withHit + memory(v) + +class MemoizedWithDefault[From, To](fun: From => To, default: To) extends Memoized[From, To](fun): + override def handleFault(): To = default + +def memoized[A, B](f: A => B): A => B = Memoized(f) +def memoized[A, B](f: A => B, default: B): A => B = MemoizedWithDefault(f, default) diff --git a/proofs/prooftrace2.names b/proofs/prooftrace2.names new file mode 100644 index 000000000..ca88089d2 --- /dev/null +++ b/proofs/prooftrace2.names @@ -0,0 +1,70 @@ +[{"id": 1838, "nm": "ABS_SIMP"}, +{"id": 6189, "nm": "AND_CLAUSES"}, +{"id": 18, "nm": "AND_DEF"}, +{"id": 15458, "nm": "AND_FORALL_THM"}, +{"id": 1753, "nm": "BETA_THM"}, +{"id": 2472, "nm": "CONJ_ACI"}, +{"id": 1982, "nm": "CONJ_ASSOC"}, +{"id": 2070, "nm": "CONJ_SYM"}, +{"id": 3526, "nm": "DISJ_ACI"}, +{"id": 2702, "nm": "DISJ_ASSOC"}, +{"id": 2836, "nm": "DISJ_SYM"}, +{"id": 5775, "nm": "EQ_CLAUSES"}, +{"id": 1489, "nm": "EQ_REFL"}, +{"id": 1558, "nm": "EQ_SYM"}, +{"id": 1631, "nm": "EQ_SYM_EQ"}, +{"id": 1716, "nm": "EQ_TRANS"}, +{"id": 463, "nm": "EXISTS_DEF"}, +{"id": 16337, "nm": "EXISTS_OR_THM"}, +{"id": 13378, "nm": "EXISTS_REFL"}, +{"id": 4748, "nm": "EXISTS_SIMP"}, +{"id": 27262, "nm": "EXISTS_UNIQUE"}, +{"id": 25242, "nm": "EXISTS_UNIQUE_ALT"}, +{"id": 830, "nm": "EXISTS_UNIQUE_DEF"}, +{"id": 13591, "nm": "EXISTS_UNIQUE_REFL"}, +{"id": 13353, "nm": "EXISTS_UNIQUE_THM"}, +{"id": 15077, "nm": "FORALL_AND_THM"}, +{"id": 421, "nm": "FORALL_DEF"}, +{"id": 4520, "nm": "FORALL_SIMP"}, +{"id": 14060, "nm": "FORALL_UNWIND_THM1"}, +{"id": 13982, "nm": "FORALL_UNWIND_THM2"}, +{"id": 715, "nm": "F_DEF"}, +{"id": 6987, "nm": "IMP_CLAUSES"}, +{"id": 3722, "nm": "IMP_CONJ"}, +{"id": 3932, "nm": "IMP_CONJ_ALT"}, +{"id": 188, "nm": "IMP_DEF"}, +{"id": 18844, "nm": "LEFT_AND_EXISTS_THM"}, +{"id": 15720, "nm": "LEFT_AND_FORALL_THM"}, +{"id": 17685, "nm": "LEFT_EXISTS_AND_THM"}, +{"id": 21758, "nm": "LEFT_FORALL_IMP_THM"}, +{"id": 21428, "nm": "LEFT_IMP_EXISTS_THM"}, +{"id": 4153, "nm": "LEFT_OR_DISTRIB"}, +{"id": 16988, "nm": "LEFT_OR_EXISTS_THM"}, +{"id": 23730, "nm": "MONO_EXISTS"}, +{"id": 23541, "nm": "MONO_FORALL"}, +{"id": 5953, "nm": "NOT_CLAUSES_WEAK"}, +{"id": 716, "nm": "NOT_DEF"}, +{"id": 6495, "nm": "OR_CLAUSES"}, +{"id": 567, "nm": "OR_DEF"}, +{"id": 16692, "nm": "OR_EXISTS_THM"}, +{"id": 1516, "nm": "REFL_CLAUSE"}, +{"id": 19244, "nm": "RIGHT_AND_EXISTS_THM"}, +{"id": 15982, "nm": "RIGHT_AND_FORALL_THM"}, +{"id": 18085, "nm": "RIGHT_EXISTS_AND_THM"}, +{"id": 21098, "nm": "RIGHT_FORALL_IMP_THM"}, +{"id": 20804, "nm": "RIGHT_IMP_FORALL_THM"}, +{"id": 4374, "nm": "RIGHT_OR_DISTRIB"}, +{"id": 17284, "nm": "RIGHT_OR_EXISTS_THM"}, +{"id": 14696, "nm": "SWAP_EXISTS_THM"}, +{"id": 14384, "nm": "SWAP_FORALL_THM"}, +{"id": 19602, "nm": "TRIV_AND_EXISTS_THM"}, +{"id": 18443, "nm": "TRIV_EXISTS_AND_THM"}, +{"id": 22596, "nm": "TRIV_EXISTS_IMP_THM"}, +{"id": 22071, "nm": "TRIV_FORALL_IMP_THM"}, +{"id": 20056, "nm": "TRIV_FORALL_OR_THM"}, +{"id": 20510, "nm": "TRIV_OR_FORALL_THM"}, +{"id": 0, "nm": "T_DEF"}, +{"id": 13752, "nm": "UNWIND_THM1"}, +{"id": 13830, "nm": "UNWIND_THM2"}, +{"id": 24328, "nm": "WLOG_RELATION"}, +{"id": 867, "nm": "_FALSITY_"}] diff --git a/proofs/prooftrace2.proofs b/proofs/prooftrace2.proofs new file mode 100644 index 000000000..95b2bd07e --- /dev/null +++ b/proofs/prooftrace2.proofs @@ -0,0 +1,27305 @@ +[ +{"id": 0, "pr": {"name": "DEFINITION", "dep1": 0, "dep2": 0, "strdep": "T", "termdep": "C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(c(T)(T[bool][])))(C(C(c(=)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]]]))(A(v(p)(T[bool][]))(v(p)(T[bool][]))))(A(v(p)(T[bool][]))(v(p)(T[bool][]))))", "termsdeps": [], "typesdeps": []}}, +{"id": 1, "pr": {"name": "REFL", "dep1": 0, "dep2": 0, "strdep": "", "termdep": "A(v(p)(T[bool][]))(v(p)(T[bool][]))", "termsdeps": [], "typesdeps": []}}, +{"id": 2, "pr": {"name": "REFL", "dep1": 0, "dep2": 0, "strdep": "", "termdep": "c(T)(T[bool][])", "termsdeps": [], "typesdeps": []}}, +{"id": 3, "pr": {"name": "REFL", "dep1": 0, 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"C(C(c(=)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]]]))(A(v(t2)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t3)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t3)(T[bool][])))))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(v(t3)(T[bool][]))))))))(A(v(t2)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t3)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t3)(T[bool][])))))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(v(t3)(T[bool][])))))))"}}, +{"id": 1966, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]]]))(A(v(t2)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t3)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t3)(T[bool][])))))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(v(t3)(T[bool][]))))))))(A(v(t2)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t3)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t3)(T[bool][])))))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(v(t3)(T[bool][])))))))"}}, +{"id": 1967, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]]]))(A(v(t2)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t3)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t3)(T[bool][])))))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(v(t3)(T[bool][]))))))))(A(v(t2)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t3)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t3)(T[bool][])))))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(v(t3)(T[bool][])))))))"}}, +{"id": 1968, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]][T[fun][[T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]][T[bool][]]]]]))(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]])))(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))"}}, +{"id": 1969, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t3)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t3)(T[bool][])))))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(v(t3)(T[bool][])))))))))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t3)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t3)(T[bool][])))))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(v(t3)(T[bool][]))))))))"}}, +{"id": 1970, "th": {"hy": [], "cc": "C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t3)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t3)(T[bool][])))))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(v(t3)(T[bool][])))))))"}}, +{"id": 1971, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(=)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]]]))(v(P)(T[fun][[T[bool][]][T[bool][]]])))(A(v(x)(T[bool][]))(c(T)(T[bool][])))))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(v(P)(T[fun][[T[bool][]][T[bool][]]])))"}}, +{"id": 1972, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t3)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t3)(T[bool][])))))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(v(t3)(T[bool][])))))))))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t3)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t3)(T[bool][])))))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(v(t3)(T[bool][])))))))))(c(T)(T[bool][])))"}}, +{"id": 1973, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t3)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t3)(T[bool][])))))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(/\\)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(v(t3)(T[bool][])))))))))(c(T)(T[bool][]))"}}, +{"id": 1974, "th": {"hy": [], "cc": 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"C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][])))))(c(T)(T[bool][]))"}}, +{"id": 2816, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]]]))(A(v(t2)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][]))))))(A(v(t2)(T[bool][]))(c(T)(T[bool][])))"}}, +{"id": 2817, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(=)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]]]))(A(v(t2)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][]))))))(A(v(x)(T[bool][]))(c(T)(T[bool][])))))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][]))))))"}}, +{"id": 2818, "th": {"hy": [], "cc": 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"C(C(c(=)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]]]))(A(v(t2)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][]))))))(A(v(t2)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][])))))"}}, +{"id": 2820, "th": {"hy": [], "cc": 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"C(C(c(=)(T[fun][[T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]][T[fun][[T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]][T[bool][]]]]]))(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]])))(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))"}}, +{"id": 2823, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][])))))))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][]))))))"}}, +{"id": 2824, "th": {"hy": [], "cc": 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"C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][])))))))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][])))))))(c(T)(T[bool][])))"}}, +{"id": 2827, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][])))))))(c(T)(T[bool][]))"}}, +{"id": 2828, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]]]))(A(v(t1)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][]))))))))(A(v(t1)(T[bool][]))(c(T)(T[bool][])))"}}, +{"id": 2829, "th": {"hy": [], "cc": 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+{"id": 2830, "th": {"hy": [], "cc": "C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t1)(T[bool][]))(C(c(!)(T[fun][[T[fun][[T[bool][]][T[bool][]]]][T[bool][]]]))(A(v(t2)(T[bool][]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t1)(T[bool][])))(v(t2)(T[bool][]))))(C(C(c(\\/)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(v(t2)(T[bool][])))(v(t1)(T[bool][])))))))"}}, +{"id": 2831, "th": {"hy": [], "cc": 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"C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(!)(T[fun][[T[fun][[t[A]][T[bool][]]]][T[bool][]]]))(A(v(x)(t[A]))(v(t)(T[bool][])))))(v(t)(T[bool][]))))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(!)(T[fun][[T[fun][[t[A]][T[bool][]]]][T[bool][]]]))(A(v(x)(t[A]))(v(t)(T[bool][])))))(v(t)(T[bool][])))"}}, +{"id": 26938, "th": {"hy": [], "cc": "C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(!)(T[fun][[T[fun][[t[A]][T[bool][]]]][T[bool][]]]))(A(v(x)(t[A]))(v(t)(T[bool][])))))(v(t)(T[bool][]))))(C(C(c(=)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(!)(T[fun][[T[fun][[t[A]][T[bool][]]]][T[bool][]]]))(A(v(y)(t[A]))(v(t)(T[bool][])))))(v(t)(T[bool][])))"}}, +{"id": 26939, "th": {"hy": [], "cc": 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"C(C(c(==>)(T[fun][[T[bool][]][T[fun][[T[bool][]][T[bool][]]]]]))(C(c(?!)(T[fun][[T[fun][[t[A]][T[bool][]]]][T[bool][]]]))(v(P)(T[fun][[t[A]][T[bool][]]]))))(C(c(?)(T[fun][[T[fun][[t[A]][T[bool][]]]][T[bool][]]]))(v(P)(T[fun][[t[A]][T[bool][]]])))"}} +] \ No newline at end of file diff --git a/refman/lisa.pdf b/refman/lisa.pdf index e9c6913a9..7a56de5a6 100644 Binary files a/refman/lisa.pdf and b/refman/lisa.pdf differ diff --git a/refman/lisa.tex b/refman/lisa.tex index d78e27da4..585bdcd1a 100644 --- a/refman/lisa.tex +++ b/refman/lisa.tex @@ -39,7 +39,7 @@ % Uncomment once the chapter says something useful to readers as opposed to authors: %\chapter*{Introduction} -%This document aims to give a complete documentation on Lisa. Tentatively, every chapter and section will explain a part or concept of Lisa, and explains both its implementation and its theoretical foundations. +%This document aims to give a complete documentation on Lisa. Tentatively, every chapter and section will explain a part or concept of Lisa, and explains both its implementation and its theoretical foundations. sadf sredg. \tableofcontents