From a60b942bc6b859555703cc68ccabe2aacb937ec8 Mon Sep 17 00:00:00 2001 From: SimonGuilloud <simon.guilloud@epfl.ch> Date: Mon, 8 Jan 2024 12:38:13 +0100 Subject: [PATCH] Substitution bellow quantifier (#203) * Add the file CHANGES.md * tracking inconsistencies * implement a first version of substeq2 * tracking a bug * 2-version works * removed all version, renamed versions 2 to normal. All seems to work (including Recursion.scala). Haven't run tests yet, nor tested on function substitutions * All tests and the library work. Added sanity checks for substitution steps. * Add tests * scalafix, scalafmt * update CHANGES.md * "of" for quantified facts has been implemented and tested. * manual and changelist updated, order between forall and free instantiation swapped. scalafix, scalafmt. * correct typos --- CHANGES.md | 11 + lisa-examples/src/main/scala/Test.scala | 58 +++++ .../scala/lisa/kernel/fol/Substitutions.scala | 8 +- .../lisa/kernel/proof/SCProofChecker.scala | 178 ++++++++++----- .../lisa/kernel/proof/SequentCalculus.scala | 9 +- .../scala/lisa/automation/CommonTactics.scala | 2 +- .../scala/lisa/automation/Substitution.scala | 89 +++++--- .../main/scala/lisa/automation/Tableau.scala | 3 + .../scala/lisa/automation/Tautology.scala | 14 +- .../main/scala/lisa/maths/Quantifiers.scala | 4 +- .../lisa/maths/settheory/InductiveSets.scala | 2 +- .../lisa/maths/settheory/SetTheory.scala | 75 +++--- .../lisa/maths/settheory/SetTheory2.scala | 4 + .../maths/settheory/orderings/Recursion.scala | 4 - .../src/main/scala/lisa/fol/Common.scala | 1 + .../src/main/scala/lisa/fol/Sequents.scala | 33 ++- .../scala/lisa/prooflib/BasicStepTactic.scala | 165 +++++++++----- .../scala/lisa/prooflib/ProofsHelpers.scala | 2 +- .../scala/lisa/prooflib/WithTheorems.scala | 28 ++- .../main/scala/lisa/utils/Serialization.scala | 59 +++-- .../lisa/kernel/IncorrectProofsTests.scala | 10 +- .../test/scala/lisa/kernel/ProofTests.scala | 215 +++++++++++++++++- refman/kernel.tex | 49 ++-- refman/lisa.pdf | Bin 239313 -> 243597 bytes refman/macro.tex | 5 + refman/prooflib.tex | 32 ++- 26 files changed, 781 insertions(+), 279 deletions(-) create mode 100644 lisa-examples/src/main/scala/Test.scala diff --git a/CHANGES.md b/CHANGES.md index 25595ade..8907b305 100644 --- a/CHANGES.md +++ b/CHANGES.md @@ -1,5 +1,16 @@ # Change List + +## 2024-01-02 +The of keyword can now be used to instantiate quantified variables in facts whose right hand side is a single universaly quantified formula. This can be combined freely with instantiation of ffree schematic symbols. Manual and tests update. + + +## 2023-12-31 +Expanded the Substitution rules to allow substitution under quantifiers when a statement of the form $\forall x. f(x) = g(x)$ is given. + +## 2023-12-29 +Update to the user manual, change format to B5 and more improvements. + ## 2023-12-19 Introduction of local definitions, with methods `witness`, `t.replace`, `t.collect`, `t.map` and `t.filter`. Update of the manual, describing how those work, along with some corrections to the statement of axioms ot match LISA's presentation. Include test cases using each, and proved some required theorems. Fix an error in the reconstruction of OL-normalized formulas used in some tactics. diff --git a/lisa-examples/src/main/scala/Test.scala b/lisa-examples/src/main/scala/Test.scala new file mode 100644 index 00000000..b65c3c6c --- /dev/null +++ b/lisa-examples/src/main/scala/Test.scala @@ -0,0 +1,58 @@ + +object Test extends lisa.Main { + + val u = variable + val v = variable + val w = variable + val x = variable + val y = variable + val z = variable + val a = variable + val c = variable + val d = variable + + val f = function[1] + val g = function[1] + val h = function[2] + + val E = predicate[2] + val P = predicate[2] + + val assump1 = ∀(u, ∀(v, ∀(w, E(u, v) ==> (E(v, w) ==> E(u, w))))) + val assump2 = ∀(x, ∀(y, E(x, y) ==> (E(f(x), f(y))))) + val assump3 = ∀(z, E(f(g(z)), g(f(z)))) + + val goal = E(f(f(g(a))), g(f(f(a)))) + + + + val thm = Theorem((assump1, assump2, assump3) |- goal) { + have(thesis) by Tableau + } + + val thm1 = Theorem(∀(x, E(x, x)) |- ∀(x, E(f(x), f(x)))) { + val s1 = assume(∀(x, E(x, x))) + have(thesis) by RightForall(s1 of f(x)) + } + + val thm2 = Theorem(∀(y, ∀(x, E(x, y))) |- ∀(y, ∀(x, E(f(x), h(x, y))))) { + val s1 = assume(∀(y, ∀(x, E(x, y)))) + println((s1 of (h(x, y), f(x))).result) + have(∀(x, E(f(x), h(x, y)))) by RightForall(s1 of (h(x, y), f(x))) + thenHave(thesis) by RightForall + } + + val thm3 = Theorem(∀(y, ∀(x, E(x, y))) |- E(f(x), y) /\ E(x, h(x, y))) { + val s1 = assume (∀(y, ∀(x, E(x, y)))) + val s2 = have(∀(x, E(x, y))) by Restate.from(s1 of y) + have(thesis) by Tautology.from(s2 of f(x), s2 of (x, y := h(x, y))) + + } + + val ax = Axiom(∀(x, ∀(y, P(x, y)))) + val thm4 = Theorem(c === d) { + have(thesis) by Restate.from(ax of (c, d, P := ===)) + showCurrentProof() + } + +} diff --git a/lisa-kernel/src/main/scala/lisa/kernel/fol/Substitutions.scala b/lisa-kernel/src/main/scala/lisa/kernel/fol/Substitutions.scala index f3f04bc2..51bfb465 100644 --- a/lisa-kernel/src/main/scala/lisa/kernel/fol/Substitutions.scala +++ b/lisa-kernel/src/main/scala/lisa/kernel/fol/Substitutions.scala @@ -136,7 +136,7 @@ trait Substitutions extends FormulaDefinitions { case ConnectorFormula(label, args) => ConnectorFormula(label, args.map(instantiateTermSchemas(_, m))) case BinderFormula(label, bound, inner) => val newSubst = m - bound - val fv: Set[VariableLabel] = newSubst.flatMap { case (symbol, LambdaTermTerm(arguments, body)) => body.freeVariables }.toSet + val fv: Set[VariableLabel] = newSubst.flatMap { case (symbol, LambdaTermTerm(arguments, body)) => body.freeVariables }.toSet ++ inner.freeVariables if (fv.contains(bound)) { val newBoundVariable = VariableLabel(freshId(fv.map(_.name) ++ m.keys.map(_.id), bound.name)) val newInner = substituteVariablesInFormula(inner, Map(bound -> VariableTerm(newBoundVariable))) @@ -162,7 +162,7 @@ trait Substitutions extends FormulaDefinitions { } case ConnectorFormula(label, args) => ConnectorFormula(label, args.map(instantiatePredicateSchemas(_, m))) case BinderFormula(label, bound, inner) => - val fv: Set[VariableLabel] = (m.flatMap { case (symbol, LambdaTermFormula(arguments, body)) => body.freeVariables }).toSet + val fv: Set[VariableLabel] = (m.flatMap { case (symbol, LambdaTermFormula(arguments, body)) => body.freeVariables }).toSet ++ inner.freeVariables if (fv.contains(bound)) { val newBoundVariable = VariableLabel(freshId(fv.map(_.name), bound.name)) val newInner = substituteVariablesInFormula(inner, Map(bound -> VariableTerm(newBoundVariable))) @@ -189,7 +189,7 @@ trait Substitutions extends FormulaDefinitions { case _ => ConnectorFormula(label, newArgs) } case BinderFormula(label, bound, inner) => - val fv: Set[VariableLabel] = (m.flatMap { case (symbol, LambdaFormulaFormula(arguments, body)) => body.freeVariables }).toSet + val fv: Set[VariableLabel] = (m.flatMap { case (symbol, LambdaFormulaFormula(arguments, body)) => body.freeVariables }).toSet ++ inner.freeVariables if (fv.contains(bound)) { val newBoundVariable = VariableLabel(freshId(fv.map(_.name), bound.name)) val newInner = substituteVariablesInFormula(inner, Map(bound -> VariableTerm(newBoundVariable))) @@ -232,7 +232,7 @@ trait Substitutions extends FormulaDefinitions { val fv: Set[VariableLabel] = (mCon.flatMap { case (symbol, LambdaFormulaFormula(arguments, body)) => body.freeVariables }).toSet ++ (mPred.flatMap { case (symbol, LambdaTermFormula(arguments, body)) => body.freeVariables }).toSet ++ - (mTerm.flatMap { case (symbol, LambdaTermTerm(arguments, body)) => body.freeVariables }).toSet + (mTerm.flatMap { case (symbol, LambdaTermTerm(arguments, body)) => body.freeVariables }).toSet ++ inner.freeVariables if (fv.contains(bound)) { val newBoundVariable = VariableLabel(freshId(fv.map(_.name) ++ mTerm.keys.map(_.id), bound.name)) val newInner = substituteVariablesInFormula(inner, Map(bound -> VariableTerm(newBoundVariable))) diff --git a/lisa-kernel/src/main/scala/lisa/kernel/proof/SCProofChecker.scala b/lisa-kernel/src/main/scala/lisa/kernel/proof/SCProofChecker.scala index a4c89b54..bf7c7114 100644 --- a/lisa-kernel/src/main/scala/lisa/kernel/proof/SCProofChecker.scala +++ b/lisa-kernel/src/main/scala/lisa/kernel/proof/SCProofChecker.scala @@ -356,23 +356,35 @@ object SCProofChecker { */ case LeftSubstEq(b, t1, equals, lambdaPhi) => val (s_es, t_es) = equals.unzip - val phi_s_for_f = lambdaPhi(s_es) - val phi_t_for_f = lambdaPhi(t_es) - val sEqT_es = equals map { case (s, t) => AtomicFormula(equality, Seq(s, t)) } + val (phi_args, phi_body) = lambdaPhi + if (phi_args.size != s_es.size) // Not strictly necessary, but it's a good sanity check. To reactivate when tactics have been modified. + SCInvalidProof(SCProof(step), Nil, "The number of arguments of φ must be the same as the number of equalities.") + else if (equals.zip(phi_args).exists { case ((s, t), arg) => s.vars.size != arg.arity || t.vars.size != arg.arity }) + SCInvalidProof(SCProof(step), Nil, "The arities of symbols in φ must be the same as the arities of equalities.") + else { + val phi_s_for_f = instantiateTermSchemas(phi_body, (phi_args zip s_es).toMap) + val phi_t_for_f = instantiateTermSchemas(phi_body, (phi_args zip t_es).toMap) + val sEqT_es = equals map { + case (s, t) => + assert(s.vars.size == t.vars.size) + val base = AtomicFormula(equality, Seq(s.body, if (s.vars == t.vars) t.body else t(s.vars.map(VariableTerm)))) + (s.vars).foldLeft(base: Formula) { case (acc, s_arg) => BinderFormula(Forall, s_arg, acc) } + } - if (isSameSet(b.right, ref(t1).right)) - if ( - isSameSet(b.left + phi_t_for_f, ref(t1).left ++ sEqT_es + phi_s_for_f) || - isSameSet(b.left + phi_s_for_f, ref(t1).left ++ sEqT_es + phi_t_for_f) - ) - SCValidProof(SCProof(step)) - else - SCInvalidProof( - SCProof(step), - Nil, - "Left-hand sides of the conclusion + φ(s_) must be the same as left-hand side of the premise + (s=t)_ + φ(t_) (or with s_ and t_ swapped)." + if (isSameSet(b.right, ref(t1).right)) + if ( + isSameSet(b.left + phi_t_for_f, ref(t1).left ++ sEqT_es + phi_s_for_f) || + isSameSet(b.left + phi_s_for_f, ref(t1).left ++ sEqT_es + phi_t_for_f) ) - else SCInvalidProof(SCProof(step), Nil, "Right-hand sides of the premise and the conclusion aren't the same.") + SCValidProof(SCProof(step)) + else + SCInvalidProof( + SCProof(step), + Nil, + "Left-hand sides of the conclusion + φ(s_) must be the same as left-hand side of the premise + (s=t)_ + φ(t_) (or with s_ and t_ swapped)." + ) + else SCInvalidProof(SCProof(step), Nil, "Right-hand sides of the premise and the conclusion aren't the same.") + } /* * Γ |- φ(s_), Δ @@ -380,46 +392,73 @@ object SCProofChecker { * Γ, (s=t)_ |- φ(t_), Δ */ case RightSubstEq(b, t1, equals, lambdaPhi) => - val sEqT_es = equals map { case (s, t) => AtomicFormula(equality, Seq(s, t)) } - if (isSameSet(ref(t1).left ++ sEqT_es, b.left)) { - val (s_es, t_es) = equals.unzip - val phi_s_for_f = lambdaPhi(s_es) - val phi_t_for_f = lambdaPhi(t_es) - if ( - isSameSet(b.right + phi_s_for_f, ref(t1).right + phi_t_for_f) || - isSameSet(b.right + phi_t_for_f, ref(t1).right + phi_s_for_f) - ) - SCValidProof(SCProof(step)) - else - SCInvalidProof( - SCProof(step), - Nil, - s"Right-hand side of the premise and the conclusion should be the same with each containing one of φ(s_) φ(t_), but it isn't the case." + val (s_es, t_es) = equals.unzip + val (phi_args, phi_body) = lambdaPhi + if (phi_args.size != equals.size) // Not strictly necessary, but it's a good sanity check. To reactivate when tactics have been modified. + SCInvalidProof(SCProof(step), Nil, "The number of arguments of φ must be the same as the number of equalities.") + else if (equals.zip(phi_args).exists { case ((s, t), arg) => s.vars.size != arg.arity || t.vars.size != arg.arity }) + SCInvalidProof(SCProof(step), Nil, "The arities of symbols in φ must be the same as the arities of equalities.") + else { + val phi_s_for_f = instantiateTermSchemas(phi_body, (phi_args zip s_es).toMap) + val phi_t_for_f = instantiateTermSchemas(phi_body, (phi_args zip t_es).toMap) + val sEqT_es = equals map { + case (s, t) => + assert(s.vars.size == t.vars.size) + val base = AtomicFormula(equality, Seq(s.body, if (s.vars == t.vars) t.body else t(s.vars.map(VariableTerm)))) + (s.vars).foldLeft(base: Formula) { case (acc, s_arg) => BinderFormula(Forall, s_arg, acc) } + } + + if (isSameSet(ref(t1).left ++ sEqT_es, b.left)) + if ( + isSameSet(b.right + phi_s_for_f, ref(t1).right + phi_t_for_f) || + isSameSet(b.right + phi_t_for_f, ref(t1).right + phi_s_for_f) ) - } else SCInvalidProof(SCProof(step), Nil, "Left-hand sides of the premise + (s=t)_ must be the same as left-hand side of the premise.") + SCValidProof(SCProof(step)) + else + SCInvalidProof( + SCProof(step), + Nil, + "Right-hand side of the premise and the conclusion should be the same with each containing one of φ(s_) φ(t_), but it isn't the case." + ) + else SCInvalidProof(SCProof(step), Nil, "Left-hand sides of the premise + (s=t)_ must be the same as left-hand side of the premise.") + } + /* * Γ, φ(ψ_) |- Δ * --------------------- * Γ, ψ⇔τ, φ(τ) |- Δ */ case LeftSubstIff(b, t1, equals, lambdaPhi) => - val psiIffTau = equals map { case (psi, tau) => ConnectorFormula(Iff, Seq(psi, tau)) } val (phi_s, tau_s) = equals.unzip - val phi_tau_for_q = lambdaPhi(phi_s) - val phi_psi_for_q = lambdaPhi(tau_s) - if (isSameSet(b.right, ref(t1).right)) - if ( - isSameSet(ref(t1).left ++ psiIffTau + phi_tau_for_q, b.left + phi_psi_for_q) || - isSameSet(ref(t1).left ++ psiIffTau + phi_psi_for_q, b.left + phi_tau_for_q) - ) - SCValidProof(SCProof(step)) - else - SCInvalidProof( - SCProof(step), - Nil, - "Left-hand sides of the conclusion + φ(ψ_) must be the same as left-hand side of the premise + (ψ⇔τ)_ + φ(τ_) (or with ψ and τ swapped)." + val (phi_args, phi_body) = lambdaPhi + if (phi_args.size != phi_s.size) // Not strictly necessary, but it's a good sanity check. To reactivate when tactics have been modified. + SCInvalidProof(SCProof(step), Nil, "The number of arguments of φ must be the same as the number of equalities.") + else if (equals.zip(phi_args).exists { case ((s, t), arg) => s.vars.size != arg.arity || t.vars.size != arg.arity }) + SCInvalidProof(SCProof(step), Nil, "The arities of symbols in φ must be the same as the arities of equalities.") + else { + val phi_psi_for_q = instantiatePredicateSchemas(phi_body, (phi_args zip phi_s).toMap) + val phi_tau_for_q = instantiatePredicateSchemas(phi_body, (phi_args zip tau_s).toMap) + val psiIffTau = equals map { + case (s, t) => + assert(s.vars.size == t.vars.size) + val base = ConnectorFormula(Iff, Seq(s.body, if (s.vars == t.vars) t.body else t(s.vars.map(VariableTerm)))) + (s.vars).foldLeft(base: Formula) { case (acc, s_arg) => BinderFormula(Forall, s_arg, acc) } + } + + if (isSameSet(b.right, ref(t1).right)) + if ( + isSameSet(b.left + phi_tau_for_q, ref(t1).left ++ psiIffTau + phi_psi_for_q) || + isSameSet(b.left + phi_psi_for_q, ref(t1).left ++ psiIffTau + phi_tau_for_q) ) - else SCInvalidProof(SCProof(step), Nil, "Right-hand sides of the premise and the conclusion aren't the same.") + SCValidProof(SCProof(step)) + else + SCInvalidProof( + SCProof(step), + Nil, + "Left-hand sides of the conclusion + φ(ψ_) must be the same as left-hand side of the premise + (ψ⇔τ)_ + φ(τ_) (or with ψ and τ swapped)." + ) + else SCInvalidProof(SCProof(step), Nil, "Right-hand sides of the premise and the conclusion aren't the same.") + } /* * Γ |- φ[ψ/?p], Δ @@ -427,23 +466,38 @@ object SCProofChecker { * Γ, ψ⇔τ |- φ[τ/?p], Δ */ case RightSubstIff(b, t1, equals, lambdaPhi) => - val psiIffTau = equals map { case (psi, tau) => ConnectorFormula(Iff, Seq(psi, tau)) } - val (phi_s, tau_s) = equals.unzip - val phi_tau_for_q = lambdaPhi(phi_s) - val phi_psi_for_q = lambdaPhi(tau_s) - if (isSameSet(ref(t1).left ++ psiIffTau, b.left)) - if ( - isSameSet(b.right + phi_tau_for_q, ref(t1).right + phi_psi_for_q) || - isSameSet(b.right + phi_psi_for_q, ref(t1).right + phi_tau_for_q) - ) - SCValidProof(SCProof(step)) - else - SCInvalidProof( - SCProof(step), - Nil, - s"Right-hand side of the premise and the conclusion should be the same with each containing one of φ[τ/?q] and φ[ψ/?q], but it isn't the case." + val (psi_s, tau_s) = equals.unzip + val (phi_args, phi_body) = lambdaPhi + if (phi_args.size != psi_s.size) + SCInvalidProof(SCProof(step), Nil, "The number of arguments of φ must be the same as the number of equalities.") + else if (equals.zip(phi_args).exists { case ((s, t), arg) => s.vars.size != arg.arity || t.vars.size != arg.arity }) + SCInvalidProof(SCProof(step), Nil, "The arities of symbols in φ must be the same as the arities of equalities.") + else { + val phi_psi_for_q = instantiatePredicateSchemas(phi_body, (phi_args zip psi_s).toMap) + val phi_tau_for_q = instantiatePredicateSchemas(phi_body, (phi_args zip tau_s).toMap) + val psiIffTau = equals map { + case (s, t) => + assert(s.vars.size == t.vars.size) + val base = ConnectorFormula(Iff, Seq(s.body, if (s.vars == t.vars) t.body else t(s.vars.map(VariableTerm)))) + (s.vars).foldLeft(base: Formula) { case (acc, s_arg) => BinderFormula(Forall, s_arg, acc) } + } + + if (isSameSet(ref(t1).left ++ psiIffTau, b.left)) + if ( + isSameSet(b.right + phi_tau_for_q, ref(t1).right + phi_psi_for_q) || + isSameSet(b.right + phi_psi_for_q, ref(t1).right + phi_tau_for_q) ) - else SCInvalidProof(SCProof(step), Nil, "Left-hand sides of the premise + ψ⇔τ must be the same as left-hand side of the premise.") + SCValidProof(SCProof(step)) + else + SCInvalidProof( + SCProof(step), + Nil, + "Right-hand side of the premise and the conclusion should be the same with each containing one of φ[τ/?q] and φ[ψ/?q], but it isn't the case." + ) + else SCInvalidProof(SCProof(step), Nil, "Left-hand sides of the premise + ψ⇔τ must be the same as left-hand side of the premise.") + } + + /** * <pre> diff --git a/lisa-kernel/src/main/scala/lisa/kernel/proof/SequentCalculus.scala b/lisa-kernel/src/main/scala/lisa/kernel/proof/SequentCalculus.scala index d089f29a..b84d84b4 100644 --- a/lisa-kernel/src/main/scala/lisa/kernel/proof/SequentCalculus.scala +++ b/lisa-kernel/src/main/scala/lisa/kernel/proof/SequentCalculus.scala @@ -282,7 +282,7 @@ object SequentCalculus { * Γ, s1=t1, ..., sn=tn, φ(t1,...tn) |- Δ * </pre> */ - case class LeftSubstEq(bot: Sequent, t1: Int, equals: List[(Term, Term)], lambdaPhi: LambdaTermFormula) extends SCProofStep { val premises = Seq(t1) } + case class LeftSubstEq(bot: Sequent, t1: Int, equals: List[(LambdaTermTerm, LambdaTermTerm)], lambdaPhi: (Seq[SchematicTermLabel], Formula)) extends SCProofStep { val premises = Seq(t1) } /** * <pre> @@ -291,7 +291,7 @@ object SequentCalculus { * Γ, s1=t1, ..., sn=tn |- φ(t1,...tn), Δ * </pre> */ - case class RightSubstEq(bot: Sequent, t1: Int, equals: List[(Term, Term)], lambdaPhi: LambdaTermFormula) extends SCProofStep { val premises = Seq(t1) } + case class RightSubstEq(bot: Sequent, t1: Int, equals: List[(LambdaTermTerm, LambdaTermTerm)], lambdaPhi: (Seq[SchematicTermLabel], Formula)) extends SCProofStep { val premises = Seq(t1) } /** * <pre> @@ -300,7 +300,7 @@ object SequentCalculus { * Γ, a1⇔b1, ..., an⇔bn, φ(b1,...bn) |- Δ * </pre> */ - case class LeftSubstIff(bot: Sequent, t1: Int, equals: List[(Formula, Formula)], lambdaPhi: LambdaFormulaFormula) extends SCProofStep { val premises = Seq(t1) } + case class LeftSubstIff(bot: Sequent, t1: Int, equals: List[(LambdaTermFormula, LambdaTermFormula)], lambdaPhi: (Seq[SchematicAtomicLabel], Formula)) extends SCProofStep { val premises = Seq(t1) } /** * <pre> @@ -309,7 +309,8 @@ object SequentCalculus { * Γ, a1⇔b1, ..., an⇔bn |- φ(b1,...bn), Δ * </pre> */ - case class RightSubstIff(bot: Sequent, t1: Int, equals: List[(Formula, Formula)], lambdaPhi: LambdaFormulaFormula) extends SCProofStep { val premises = Seq(t1) } + + case class RightSubstIff(bot: Sequent, t1: Int, equals: List[(LambdaTermFormula, LambdaTermFormula)], lambdaPhi: (Seq[SchematicAtomicLabel], Formula)) extends SCProofStep { val premises = Seq(t1) } // Rule for schemas diff --git a/lisa-sets/src/main/scala/lisa/automation/CommonTactics.scala b/lisa-sets/src/main/scala/lisa/automation/CommonTactics.scala index 383ed4f8..9430195a 100644 --- a/lisa-sets/src/main/scala/lisa/automation/CommonTactics.scala +++ b/lisa-sets/src/main/scala/lisa/automation/CommonTactics.scala @@ -65,7 +65,7 @@ object CommonTactics { val forward = lib.have(phi |- ((x === y) ==> substPhi)) subproof { lib.assume(phi) - lib.thenHave((x === y) |- substPhi) by RightSubstEq.withParameters(List((x, y)), F.lambda(x, phi)) + lib.thenHave((x === y) |- substPhi) by RightSubstEq.withParametersSimple(List((x, y)), F.lambda(x, phi)) lib.thenHave((x === y) ==> substPhi) by Restate } diff --git a/lisa-sets/src/main/scala/lisa/automation/Substitution.scala b/lisa-sets/src/main/scala/lisa/automation/Substitution.scala index fc7a0e2f..fcc45642 100644 --- a/lisa-sets/src/main/scala/lisa/automation/Substitution.scala +++ b/lisa-sets/src/main/scala/lisa/automation/Substitution.scala @@ -154,27 +154,39 @@ object Substitution { takenFormulaVars ) - lazy val rightPairs = premiseSequent.right zip premiseSequent.right.map(x => bot.right.find(y => UnificationUtils.getContextFormula( - x, - y, - freeEqualities, - freeIffs, - confinedEqualities, - takenTermVars, - confinedIffs, - takenFormulaVars - ).isDefined)) + lazy val rightPairs = premiseSequent.right zip premiseSequent.right.map(x => + bot.right.find(y => + UnificationUtils + .getContextFormula( + x, + y, + freeEqualities, + freeIffs, + confinedEqualities, + takenTermVars, + confinedIffs, + takenFormulaVars + ) + .isDefined + ) + ) - lazy val leftPairs = filteredPrem zip filteredPrem.map(x => filteredBot.find(y => UnificationUtils.getContextFormula( - x, - y, - freeEqualities, - freeIffs, - confinedEqualities, - takenTermVars, - confinedIffs, - takenFormulaVars - ).isDefined)) + lazy val leftPairs = filteredPrem zip filteredPrem.map(x => + filteredBot.find(y => + UnificationUtils + .getContextFormula( + x, + y, + freeEqualities, + freeIffs, + confinedEqualities, + takenTermVars, + confinedIffs, + takenFormulaVars + ) + .isDefined + ) + ) lazy val violatingFormulaLeft = leftPairs.find(_._2.isEmpty) lazy val violatingFormulaRight = rightPairs.find(_._2.isEmpty) @@ -308,7 +320,7 @@ object Substitution { val sequentAfterEq = sequentAfterEqPre ++<< (eqs |- ()) ++<< (iffs |- ()) // this uses the "lambda" (λx. Λp. body) (p = left formulas) - lib.thenHave(sequentAfterEq) by BasicStepTactic.LeftSubstEq.withParameters(termInputs.toList, lambda(termVars, ctx.body.substituteUnsafe2(formSubstL))) + lib.thenHave(sequentAfterEq) by BasicStepTactic.LeftSubstEq.withParametersSimple(termInputs.toList, lambda(termVars, ctx.body.substituteUnsafe2(formSubstL))) // left <=> val formSubstR = Map((formulaVars zip formulaInputsR)*) @@ -317,7 +329,7 @@ object Substitution { val sequentAfterIff = sequentAfterIffPre ++<< (eqs |- ()) ++<< (iffs |- ()) // this uses the "lambda" (λx. Λp. body) (x = right terms) - lib.thenHave(sequentAfterIff) by BasicStepTactic.LeftSubstIff(formulaInputs.toList, lambda(formulaVars, ctx.body.substituteUnsafe2(eqSubst))) + lib.thenHave(sequentAfterIff) by BasicStepTactic.LeftSubstIff.withParametersSimple(formulaInputs.toList, lambda(formulaVars, ctx.body.substituteUnsafe2(eqSubst))) sequentAfterIff } @@ -365,7 +377,7 @@ object Substitution { val sequentAfterEq = sequentAfterEqPre ++<< (eqs |- ()) ++<< (iffs |- ()) // this uses the "lambda" (λx. Λp. body) (p = left formulas) - lib.thenHave(sequentAfterEq) by BasicStepTactic.RightSubstEq.withParameters(termInputs.toList, lambda(termVars, ctx.body.substituteUnsafe2(formSubstL))) + lib.thenHave(sequentAfterEq) by BasicStepTactic.RightSubstEq.withParametersSimple(termInputs.toList, lambda(termVars, ctx.body.substituteUnsafe2(formSubstL))) // right <=> val formSubstR = Map((formulaVars zip formulaInputsR)*) @@ -374,7 +386,7 @@ object Substitution { val sequentAfterIff = sequentAfterIffPre ++<< (eqs |- ()) ++<< (iffs |- ()) // this uses the "lambda" (λx. Λp. body) (x = right terms) - lib.thenHave(sequentAfterIff) by BasicStepTactic.RightSubstIff(formulaInputs.toList, lambda(formulaVars, ctx.body.substituteUnsafe2(eqSubst))) + lib.thenHave(sequentAfterIff) by BasicStepTactic.RightSubstIff.withParametersSimple(formulaInputs.toList, lambda(formulaVars, ctx.body.substituteUnsafe2(eqSubst))) } // discharge any assumptions @@ -519,9 +531,19 @@ object Substitution { val result2: Sequent = result1.left |- AppliedConnector(Or, newright.toSeq) var scproof: Seq[K.SCProofStep] = Seq(K.Restate((leftOrigin |- rightOrigin).underlying, -1)) if (toLeft) - scproof = scproof :+ K.LeftSubstEq(result1.underlying, scproof.length - 1, List(left.underlying -> right.underlying), K.LambdaTermFormula(Seq(v.underlyingLabel), leftForm.underlying)) + scproof = scproof :+ K.LeftSubstEq( + result1.underlying, + scproof.length - 1, + List(K.LambdaTermTerm(Seq(), left.underlying) -> (K.LambdaTermTerm(Seq(), right.underlying))), + (Seq(v.underlyingLabel), leftForm.underlying) + ) if (toRight) - scproof = scproof :+ K.RightSubstEq(result2.underlying, scproof.length - 1, List(left.underlying -> right.underlying), K.LambdaTermFormula(Seq(v.underlyingLabel), rightForm.underlying)) + scproof = scproof :+ K.RightSubstEq( + result2.underlying, + scproof.length - 1, + List(K.LambdaTermTerm(Seq(), left.underlying) -> (K.LambdaTermTerm(Seq(), right.underlying))), + (Seq(v.underlyingLabel), rightForm.underlying) + ) val bot = newleft + phi |- newright scproof = scproof :+ K.Restate(bot.underlying, scproof.length - 1) @@ -556,10 +578,19 @@ object Substitution { var scproof: Seq[K.SCProofStep] = Seq(K.Restate((leftOrigin |- rightOrigin).underlying, -1)) if (toLeft) - scproof = scproof :+ K.LeftSubstIff(result1.underlying, scproof.length - 1, List(left.underlying -> right.underlying), K.LambdaFormulaFormula(Seq(H.underlyingLabel), leftForm.underlying)) + scproof = scproof :+ K.LeftSubstIff( + result1.underlying, + scproof.length - 1, + List(K.LambdaTermFormula(Seq(), left.underlying) -> (K.LambdaTermFormula(Seq(), right.underlying))), + (Seq(H.underlyingLabel), leftForm.underlying) + ) if (toRight) - scproof = - scproof :+ K.RightSubstIff(result2.underlying, scproof.length - 1, List(left.underlying -> right.underlying), K.LambdaFormulaFormula(Seq(H.underlyingLabel), rightForm.underlying)) + scproof = scproof :+ K.RightSubstIff( + result2.underlying, + scproof.length - 1, + List(K.LambdaTermFormula(Seq(), left.underlying) -> (K.LambdaTermFormula(Seq(), right.underlying))), + (Seq(H.underlyingLabel), rightForm.underlying) + ) val bot = newleft + phi |- newright scproof = scproof :+ K.Restate(bot.underlying, scproof.length - 1) diff --git a/lisa-sets/src/main/scala/lisa/automation/Tableau.scala b/lisa-sets/src/main/scala/lisa/automation/Tableau.scala index 1ab6e067..de6f6bcb 100644 --- a/lisa-sets/src/main/scala/lisa/automation/Tableau.scala +++ b/lisa-sets/src/main/scala/lisa/automation/Tableau.scala @@ -24,6 +24,9 @@ import scala.collection.immutable.HashSet object Tableau extends ProofTactic with ProofSequentTactic with ProofFactSequentTactic { + var debug = true + def pr(s: Object) = if debug then println(s) + def apply(using lib: Library, proof: lib.Proof)(bot: F.Sequent): proof.ProofTacticJudgement = { solve(bot) match { case Some(value) => proof.ValidProofTactic(bot, value.steps, Seq()) diff --git a/lisa-sets/src/main/scala/lisa/automation/Tautology.scala b/lisa-sets/src/main/scala/lisa/automation/Tautology.scala index f217058c..476df8a5 100644 --- a/lisa-sets/src/main/scala/lisa/automation/Tautology.scala +++ b/lisa-sets/src/main/scala/lisa/automation/Tautology.scala @@ -144,10 +144,20 @@ object Tautology extends ProofTactic with ProofSequentTactic with ProofFactSeque val seq1 = AugSequent((atom :: s.decisions._1, s.decisions._2), lambdaF(Seq(top()))) val proof1 = solveAugSequent(seq1, offset) - val subst1 = RightSubstIff(atom :: s.decisions._1 ++ s.decisions._2.map((f: Formula) => Neg(f)) |- redF, offset + proof1.length - 1, List((atom, top())), lambdaF) + val subst1 = RightSubstIff( + atom :: s.decisions._1 ++ s.decisions._2.map((f: Formula) => Neg(f)) |- redF, + offset + proof1.length - 1, + List((LambdaTermFormula(Seq(), atom), LambdaTermFormula(Seq(), top()))), + (lambdaF.vars, lambdaF.body) + ) val seq2 = AugSequent((s.decisions._1, atom :: s.decisions._2), lambdaF(Seq(bot()))) val proof2 = solveAugSequent(seq2, offset + proof1.length + 1) - val subst2 = RightSubstIff(Neg(atom) :: s.decisions._1 ++ s.decisions._2.map((f: Formula) => Neg(f)) |- redF, offset + proof1.length + proof2.length - 1 + 1, List((atom, bot())), lambdaF) + val subst2 = RightSubstIff( + Neg(atom) :: s.decisions._1 ++ s.decisions._2.map((f: Formula) => Neg(f)) |- redF, + offset + proof1.length + proof2.length - 1 + 1, + List((LambdaTermFormula(Seq(), atom), LambdaTermFormula(Seq(), bot()))), + (lambdaF.vars, lambdaF.body) + ) val red2 = Restate(s.decisions._1 ++ s.decisions._2.map((f: Formula) => Neg(f)) |- (redF, atom), offset + proof1.length + proof2.length + 2 - 1) val cutStep = Cut(s.decisions._1 ++ s.decisions._2.map((f: Formula) => Neg(f)) |- redF, offset + proof1.length + proof2.length + 3 - 1, offset + proof1.length + 1 - 1, atom) val redStep = Restate(s.decisions._1 ++ s.decisions._2.map((f: Formula) => Neg(f)) |- s.formula, offset + proof1.length + proof2.length + 4 - 1) diff --git a/lisa-sets/src/main/scala/lisa/maths/Quantifiers.scala b/lisa-sets/src/main/scala/lisa/maths/Quantifiers.scala index 67a1e6aa..1aeaf55a 100644 --- a/lisa-sets/src/main/scala/lisa/maths/Quantifiers.scala +++ b/lisa-sets/src/main/scala/lisa/maths/Quantifiers.scala @@ -55,7 +55,7 @@ object Quantifiers extends lisa.Main { (x === y) /\ (y === z) |- (x === z) ) { have((x === y) |- (x === y)) by Hypothesis - thenHave(((x === y), (y === z)) |- (x === z)) by RightSubstEq.apply2 + thenHave(((x === y), (y === z)) |- (x === z)) by RightSubstEq.withParametersSimple(List((y, z)), lambda(y, x === y)) thenHave(thesis) by Restate } @@ -95,7 +95,7 @@ object Quantifiers extends lisa.Main { ) { have(exists(x, P(x) /\ (y === x)) |- P(y)) subproof { have(P(x) |- P(x)) by Hypothesis - thenHave((P(x), y === x) |- P(y)) by RightSubstEq.withParameters(List((y, x)), lambda(y, P(y))) + thenHave((P(x), y === x) |- P(y)) by RightSubstEq.withParametersSimple(List((y, x)), lambda(y, P(y))) thenHave(P(x) /\ (y === x) |- P(y)) by Restate thenHave(thesis) by LeftExists } diff --git a/lisa-sets/src/main/scala/lisa/maths/settheory/InductiveSets.scala b/lisa-sets/src/main/scala/lisa/maths/settheory/InductiveSets.scala index 996d55ee..cbc135a1 100644 --- a/lisa-sets/src/main/scala/lisa/maths/settheory/InductiveSets.scala +++ b/lisa-sets/src/main/scala/lisa/maths/settheory/InductiveSets.scala @@ -115,7 +115,7 @@ object InductiveSets extends lisa.Main { val form = formulaVariable val inductIff = thenHave( (∀(t, in(t, z) <=> (∀(x, inductive(x) ==> in(t, x)))), inductive(z) <=> (in(∅, z) /\ ∀(y, in(y, z) ==> in(successor(y), z)))) |- inductive(z) - ) by RightSubstIff(List((inductive(z), in(∅, z) /\ ∀(y, in(y, z) ==> in(successor(y), z)))), lambda(form, form)) + ) by RightSubstIff.withParametersSimple(List((inductive(z), in(∅, z) /\ ∀(y, in(y, z) ==> in(successor(y), z)))), lambda(form, form)) val inductDef = have(inductive(z) <=> (in(∅, z) /\ ∀(y, in(y, z) ==> in(successor(y), z)))) by InstFunSchema(Map(x -> z))(inductive.definition) diff --git a/lisa-sets/src/main/scala/lisa/maths/settheory/SetTheory.scala b/lisa-sets/src/main/scala/lisa/maths/settheory/SetTheory.scala index 34737dbb..069d23d7 100644 --- a/lisa-sets/src/main/scala/lisa/maths/settheory/SetTheory.scala +++ b/lisa-sets/src/main/scala/lisa/maths/settheory/SetTheory.scala @@ -97,7 +97,7 @@ object SetTheory extends lisa.Main { // forward direction have(fprop(z) |- fprop(z)) by Hypothesis thenHave(fprop(z) /\ (z === a) |- fprop(z)) by Weakening - thenHave((fprop(z) /\ (z === a), (z === a)) |- fprop(a)) by RightSubstEq.withParameters(List((z, a)), lambda(Seq(z), fprop(z))) + thenHave((fprop(z) /\ (z === a), (z === a)) |- fprop(a)) by RightSubstEq.withParametersSimple(List((z, a)), lambda(Seq(z), fprop(z))) val forward = thenHave(fprop(z) |- (z === a) ==> fprop(a)) by Restate // backward direction @@ -177,10 +177,10 @@ object SetTheory extends lisa.Main { val fwd = have(∃(a, in(z, a) /\ in(a, unorderedPair(x, y))) ==> (in(z, x) \/ in(z, y))) subproof { have((in(z, a), a === x) |- in(z, a)) by Hypothesis - val tax = thenHave((in(z, a), a === x) |- in(z, x)) by RightSubstEq.withParameters(List((a, x)), lambda(a, in(z, a))) + val tax = thenHave((in(z, a), a === x) |- in(z, x)) by RightSubstEq.withParametersSimple(List((a, x)), lambda(a, in(z, a))) have((in(z, a), a === y) |- in(z, a)) by Hypothesis - val tay = thenHave((in(z, a), a === y) |- in(z, y)) by RightSubstEq.withParameters(List((a, y)), lambda(a, in(z, a))) + val tay = thenHave((in(z, a), a === y) |- in(z, y)) by RightSubstEq.withParametersSimple(List((a, y)), lambda(a, in(z, a))) have((in(z, a), (a === x) \/ (a === y)) |- (in(z, x), in(z, y))) by LeftOr(tax, tay) thenHave((in(z, a), in(a, unorderedPair(x, y))) |- (in(z, x), in(z, y))) by Substitution.ApplyRules(upairax) @@ -191,7 +191,7 @@ object SetTheory extends lisa.Main { val bwd = have(((in(z, x) \/ in(z, y)) ==> ∃(a, in(z, a) /\ in(a, unorderedPair(x, y))))) subproof { have((in(z, x), (a === x)) |- in(z, x)) by Hypothesis - thenHave((in(z, x), (a === x)) |- in(z, a)) by RightSubstEq.withParameters(List((a, x)), lambda(a, in(z, a))) + thenHave((in(z, x), (a === x)) |- in(z, a)) by RightSubstEq.withParametersSimple(List((a, x)), lambda(a, in(z, a))) thenHave((in(z, x), (a === x)) |- in(z, a) /\ ((a === x) \/ (a === y))) by Tautology andThen(Substitution.applySubst(upairax, false)) thenHave((in(z, x), (a === x)) |- ∃(a, in(z, a) /\ in(a, unorderedPair(x, y)))) by RightExists @@ -199,7 +199,7 @@ object SetTheory extends lisa.Main { val tax = thenHave((in(z, x)) |- ∃(a, in(z, a) /\ in(a, unorderedPair(x, y)))) by Restate have((in(z, y), (a === y)) |- in(z, y)) by Hypothesis - thenHave((in(z, y), (a === y)) |- in(z, a)) by RightSubstEq.withParameters(List((a, y)), lambda(a, in(z, a))) + thenHave((in(z, y), (a === y)) |- in(z, a)) by RightSubstEq.withParametersSimple(List((a, y)), lambda(a, in(z, a))) thenHave((in(z, y), (a === y)) |- in(z, a) /\ ((a === x) \/ (a === y))) by Tautology andThen(Substitution.applySubst(upairax, false)) thenHave((in(z, y), (a === y)) |- ∃(a, in(z, a) /\ in(a, unorderedPair(x, y)))) by RightExists @@ -260,7 +260,7 @@ object SetTheory extends lisa.Main { have((in(y, x) ==> in(union(unorderedPair(y, unorderedPair(y, y))), x)) <=> (in(y, x) ==> in(union(unorderedPair(y, unorderedPair(y, y))), x))) by Restate val succEq = thenHave( (successor(y) === union(unorderedPair(y, unorderedPair(y, y)))) |- (in(y, x) ==> in(union(unorderedPair(y, unorderedPair(y, y))), x)) <=> (in(y, x) ==> in(successor(y), x)) - ) by RightSubstEq.withParameters( + ) by RightSubstEq.withParametersSimple( List((successor(y), union(unorderedPair(y, unorderedPair(y, y))))), lambda(z, (in(y, x) ==> in(union(unorderedPair(y, unorderedPair(y, y))), x)) <=> (in(y, x) ==> in(z, x))) ) @@ -291,7 +291,7 @@ object SetTheory extends lisa.Main { ∀(y, in(y, x) ==> in(union(unorderedPair(y, unorderedPair(y, y))), x)) <=> ∀(y, in(y, x) ==> in(successor(y), x)), in(∅, x) /\ ∀(y, in(y, x) ==> in(union(unorderedPair(y, unorderedPair(y, y))), x)) ) |- in(∅, x) /\ ∀(y, in(y, x) ==> in(successor(y), x)) - ) by RightSubstIff( + ) by RightSubstIff.withParametersSimple( List((∀(y, in(y, x) ==> in(union(unorderedPair(y, unorderedPair(y, y))), x)), ∀(y, in(y, x) ==> in(successor(y), x)))), lambda(form, in(∅, x) /\ form) ) @@ -301,7 +301,7 @@ object SetTheory extends lisa.Main { ∀(y, in(y, x) ==> in(union(unorderedPair(y, unorderedPair(y, y))), x)) <=> ∀(y, in(y, x) ==> in(successor(y), x)), in(∅, x) /\ ∀(y, in(y, x) ==> in(union(unorderedPair(y, unorderedPair(y, y))), x)) ) |- inductive(x) - ) by RightSubstIff(List((inductive(x), in(∅, x) /\ ∀(y, in(y, x) ==> in(successor(y), x)))), lambda(form, form)) + ) by RightSubstIff.withParametersSimple(List((inductive(x), in(∅, x) /\ ∀(y, in(y, x) ==> in(successor(y), x)))), lambda(form, form)) val cut1 = have( ( ∀(y, in(y, x) ==> in(union(unorderedPair(y, unorderedPair(y, y))), x)) <=> ∀(y, in(y, x) ==> in(successor(y), x)), @@ -532,11 +532,17 @@ object SetTheory extends lisa.Main { // backward direction // a = c and b = d => up ab = up cd (and the other case) have(unorderedPair(a, b) === unorderedPair(a, b)) by RightRefl - thenHave((a === c, b === d) |- unorderedPair(a, b) === unorderedPair(c, d)) by RightSubstEq.withParameters(List((a, c), (b, d)), lambda(Seq(x, y), unorderedPair(a, b) === unorderedPair(x, y))) + thenHave((a === c, b === d) |- unorderedPair(a, b) === unorderedPair(c, d)) by RightSubstEq.withParametersSimple( + List((a, c), (b, d)), + lambda(Seq(x, y), unorderedPair(a, b) === unorderedPair(x, y)) + ) val lhs = thenHave(Set((a === c) /\ (b === d)) |- unorderedPair(a, b) === unorderedPair(c, d)) by Restate have(unorderedPair(a, b) === unorderedPair(b, a)) by InstFunSchema(ScalaMap(x -> a, y -> b))(unorderedPairSymmetry) - thenHave((a === d, b === c) |- (unorderedPair(a, b) === unorderedPair(c, d))) by RightSubstEq.withParameters(List((a, d), (b, c)), lambda(Seq(x, y), unorderedPair(a, b) === unorderedPair(y, x))) + thenHave((a === d, b === c) |- (unorderedPair(a, b) === unorderedPair(c, d))) by RightSubstEq.withParametersSimple( + List((a, d), (b, c)), + lambda(Seq(x, y), unorderedPair(a, b) === unorderedPair(y, x)) + ) val rhs = thenHave(Set((a === d) /\ (b === c)) |- (unorderedPair(a, b) === unorderedPair(c, d))) by Restate have((((a === d) /\ (b === c)) \/ ((a === c) /\ (b === d))) |- (unorderedPair(a, b) === unorderedPair(c, d))) by LeftOr(lhs, rhs) @@ -593,7 +599,7 @@ object SetTheory extends lisa.Main { // backward direction // x === y |- {x} === {y} have(singleton(x) === singleton(x)) by RightRefl - thenHave((x === y) |- singleton(x) === singleton(y)) by RightSubstEq.withParameters(List((x, y)), lambda(a, singleton(x) === singleton(a))) + thenHave((x === y) |- singleton(x) === singleton(y)) by RightSubstEq.withParametersSimple(List((x, y)), lambda(a, singleton(x) === singleton(a))) val bwd = thenHave((x === y) ==> (singleton(x) === singleton(y))) by Restate have((singleton(x) === singleton(y)) <=> (x === y)) by RightIff(fwd, bwd) @@ -619,9 +625,9 @@ object SetTheory extends lisa.Main { val pairAxab = have(in(z, unorderedPair(a, b)) |- (z === a) \/ (z === b)) by Tautology.from(pairAxiom of (x -> a, y -> b)) have(in(a, x) /\ in(b, x) |- in(a, x)) by Restate - val za = thenHave((in(a, x) /\ in(b, x), (z === a)) |- in(z, x)) by RightSubstEq.withParameters(List((z, a)), lambda(a, in(a, x))) + val za = thenHave((in(a, x) /\ in(b, x), (z === a)) |- in(z, x)) by RightSubstEq.withParametersSimple(List((z, a)), lambda(a, in(a, x))) have(in(a, x) /\ in(b, x) |- in(b, x)) by Restate - val zb = thenHave((in(a, x) /\ in(b, x), (z === b)) |- in(z, x)) by RightSubstEq.withParameters(List((z, b)), lambda(a, in(a, x))) + val zb = thenHave((in(a, x) /\ in(b, x), (z === b)) |- in(z, x)) by RightSubstEq.withParametersSimple(List((z, b)), lambda(a, in(a, x))) val zab = have((in(a, x) /\ in(b, x), (z === a) \/ (z === b)) |- in(z, x)) by LeftOr(za, zb) @@ -665,7 +671,7 @@ object SetTheory extends lisa.Main { // (a === c) /\ (b === d) ==> pair a b === pair c d val fwd = have(((a === c) /\ (b === d)) ==> (pair(a, b) === pair(c, d))) subproof { have((pair(a, b) === pair(a, b))) by RightRefl - thenHave(Set((a === c), (b === d)) |- (pair(a, b) === pair(c, d))) by RightSubstEq.withParameters(List((a, c), (b, d)), lambda(Seq(x, y), pair(a, b) === pair(x, y))) + thenHave(Set((a === c), (b === d)) |- (pair(a, b) === pair(c, d))) by RightSubstEq.withParametersSimple(List((a, c), (b, d)), lambda(Seq(x, y), pair(a, b) === pair(x, y))) thenHave(thesis) by Restate } @@ -680,7 +686,7 @@ object SetTheory extends lisa.Main { c ))) \/ ((unorderedPair(a, b) === singleton(c)) /\ (singleton(a) === unorderedPair(c, d)))) ) |- (((unorderedPair(a, b) === unorderedPair(c, d)) /\ (singleton(a) === singleton(c))) \/ ((unorderedPair(a, b) === singleton(c)) /\ (singleton(a) === unorderedPair(c, d)))) - ) by RightSubstIff( + ) by RightSubstIff.withParametersSimple( List( ( (unorderedPair(unorderedPair(a, b), singleton(a)) === unorderedPair(unorderedPair(c, d), singleton(c))), @@ -749,7 +755,7 @@ object SetTheory extends lisa.Main { // now we need to show that the assumption is indeed true // this requires destruction of the existential quantifier in lhs have(in(x, X) /\ ∀(z, in(z, X) ==> !in(z, x)) |- in(x, X) /\ ∀(z, in(z, X) ==> !in(z, x))) by Hypothesis - val innerRhs2 = thenHave((in(y, X) /\ ∀(z, in(z, X) ==> !in(z, y)), x === y) |- in(x, X) /\ ∀(z, in(z, X) ==> !in(z, x))) by LeftSubstEq.withParameters( + val innerRhs2 = thenHave((in(y, X) /\ ∀(z, in(z, X) ==> !in(z, y)), x === y) |- in(x, X) /\ ∀(z, in(z, X) ==> !in(z, x))) by LeftSubstEq.withParametersSimple( List((x, y)), lambda(Seq(y), in(y, X) /\ ∀(z, in(z, X) ==> !in(z, y))) ) @@ -1010,7 +1016,7 @@ object SetTheory extends lisa.Main { have((in(t, z) <=> (in(t, x) /\ ∀(y, P(y) ==> in(t, y)))) |- in(t, z) <=> (in(t, x) /\ ∀(y, P(y) ==> in(t, y)))) by Hypothesis val rhs = thenHave( Set((in(t, z) <=> (in(t, x) /\ ∀(y, P(y) ==> in(t, y)))), (∀(y, P(y) ==> in(t, y)) <=> (in(t, x) /\ ∀(y, P(y) ==> in(t, y))))) |- (in(t, z) <=> (∀(y, P(y) ==> in(t, y)))) - ) by RightSubstIff(List((∀(y, P(y) ==> in(t, y)), (in(t, x) /\ ∀(y, P(y) ==> in(t, y))))), lambda(form, in(t, z) <=> (form))) + ) by RightSubstIff.withParametersSimple(List((∀(y, P(y) ==> in(t, y)), (in(t, x) /\ ∀(y, P(y) ==> in(t, y))))), lambda(form, in(t, z) <=> (form))) have((P(x), (in(t, z) <=> (in(t, x) /\ ∀(y, P(y) ==> in(t, y))))) |- in(t, z) <=> (∀(y, P(y) ==> in(t, y)))) by Cut(lhs, rhs) thenHave((P(x), ∀(t, (in(t, z) <=> (in(t, x) /\ ∀(y, P(y) ==> in(t, y)))))) |- in(t, z) <=> (∀(y, P(y) ==> in(t, y)))) by LeftForall @@ -1458,7 +1464,7 @@ object SetTheory extends lisa.Main { // (a, b) \in x * y ⟹ a ∈ x ∧ b ∈ y val fwd = have(in(pair(a, b), cartesianProduct(x, y)) ==> (in(a, x) /\ in(b, y))) subproof { have((a === c, b === d, in(c, x) /\ in(d, y)) |- in(c, x) /\ in(d, y)) by Hypothesis - thenHave((a === c, b === d, in(c, x) /\ in(d, y)) |- in(a, x) /\ in(b, y)) by RightSubstEq.withParameters(List((a, c), (b, d)), lambda(Seq(a, b), in(a, x) /\ in(b, y))) + thenHave((a === c, b === d, in(c, x) /\ in(d, y)) |- in(a, x) /\ in(b, y)) by RightSubstEq.withParametersSimple(List((a, c), (b, d)), lambda(Seq(a, b), in(a, x) /\ in(b, y))) thenHave(Set((a === c) /\ (b === d), in(c, x) /\ in(d, y)) |- in(a, x) /\ in(b, y)) by Restate andThen(Substitution.applySubst(pairExtensionality)) thenHave((pair(a, b) === pair(c, d)) /\ in(c, x) /\ in(d, y) |- in(a, x) /\ in(b, y)) by Restate @@ -1485,10 +1491,16 @@ object SetTheory extends lisa.Main { have(prem |- in(unorderedPair(a, b), powerSet(setUnion(x, y)))) by Weakening(unorderedPairInPowerSet of (x -> setUnion(x, y))) val zab = - thenHave((prem, (z === unorderedPair(a, b))) |- in(z, powerSet(setUnion(x, y)))) by RightSubstEq.withParameters(List((z, unorderedPair(a, b))), lambda(a, in(a, powerSet(setUnion(x, y))))) + thenHave((prem, (z === unorderedPair(a, b))) |- in(z, powerSet(setUnion(x, y)))) by RightSubstEq.withParametersSimple( + List((z, unorderedPair(a, b))), + lambda(a, in(a, powerSet(setUnion(x, y)))) + ) have(prem |- in(unorderedPair(a, a), powerSet(setUnion(x, y)))) by Weakening(unorderedPairInPowerSet of (x -> setUnion(x, y), b -> a)) val zaa = - thenHave((prem, (z === unorderedPair(a, a))) |- in(z, powerSet(setUnion(x, y)))) by RightSubstEq.withParameters(List((z, unorderedPair(a, a))), lambda(a, in(a, powerSet(setUnion(x, y))))) + thenHave((prem, (z === unorderedPair(a, a))) |- in(z, powerSet(setUnion(x, y)))) by RightSubstEq.withParametersSimple( + List((z, unorderedPair(a, a))), + lambda(a, in(a, powerSet(setUnion(x, y)))) + ) val cutRhs = have((prem, (z === unorderedPair(a, b)) \/ (z === singleton(a))) |- in(z, powerSet(setUnion(x, y)))) by LeftOr(zab, zaa) @@ -1849,7 +1861,7 @@ object SetTheory extends lisa.Main { a, in(pair(t, a), r) )) - ) by RightSubstIff(List((∃(a, in(pair(t, a), r)), in(t, union(union(r))) /\ ∃(a, in(pair(t, a), r)))), lambda(h, in(t, z) <=> h)) + ) by RightSubstIff.withParametersSimple(List((∃(a, in(pair(t, a), r)), in(t, union(union(r))) /\ ∃(a, in(pair(t, a), r)))), lambda(h, in(t, z) <=> h)) have((in(t, z) <=> (in(t, union(union(r))) /\ ∃(a, in(pair(t, a), r)))) |- in(t, z) <=> (∃(a, in(pair(t, a), r)))) by Cut(subst, cutRhs) thenHave(∀(t, in(t, z) <=> (in(t, union(union(r))) /\ ∃(a, in(pair(t, a), r)))) |- in(t, z) <=> (∃(a, in(pair(t, a), r)))) by LeftForall thenHave(∀(t, in(t, z) <=> (in(t, union(union(r))) /\ ∃(a, in(pair(t, a), r)))) |- ∀(t, in(t, z) <=> (∃(a, in(pair(t, a), r))))) by RightForall @@ -1917,7 +1929,7 @@ object SetTheory extends lisa.Main { a, in(pair(a, t), r) )) - ) by RightSubstIff(List((∃(a, in(pair(a, t), r)), in(t, union(union(r))) /\ ∃(a, in(pair(a, t), r)))), lambda(h, in(t, z) <=> h)) + ) by RightSubstIff.withParametersSimple(List((∃(a, in(pair(a, t), r)), in(t, union(union(r))) /\ ∃(a, in(pair(a, t), r)))), lambda(h, in(t, z) <=> h)) have((in(t, z) <=> (in(t, union(union(r))) /\ ∃(a, in(pair(a, t), r)))) |- in(t, z) <=> (∃(a, in(pair(a, t), r)))) by Cut(subst, cutRhs) thenHave(∀(t, in(t, z) <=> (in(t, union(union(r))) /\ ∃(a, in(pair(a, t), r)))) |- in(t, z) <=> (∃(a, in(pair(a, t), r)))) by LeftForall thenHave(∀(t, in(t, z) <=> (in(t, union(union(r))) /\ ∃(a, in(pair(a, t), r)))) |- ∀(t, in(t, z) <=> (∃(a, in(pair(a, t), r))))) by RightForall @@ -2373,10 +2385,11 @@ object SetTheory extends lisa.Main { val lhs = have(prem /\ ((z === y) <=> in(pair(x, y), f)) |- ((z === y) <=> ((prem ==> in(pair(x, y), f)) /\ ⊤))) subproof { val iff = have(prem |- (in(pair(x, y), f)) <=> (prem ==> in(pair(x, y), f))) by Restate have(prem /\ ((z === y) <=> in(pair(x, y), f)) |- ((z === y) <=> in(pair(x, y), f))) by Restate - val subst = thenHave((prem /\ ((z === y) <=> in(pair(x, y), f)), (in(pair(x, y), f)) <=> (prem ==> in(pair(x, y), f))) |- ((z === y) <=> (prem ==> in(pair(x, y), f)))) by RightSubstIff( - List(((in(pair(x, y), f)), (prem ==> in(pair(x, y), f)))), - lambda(h, ((z === y) <=> h)) - ) + val subst = thenHave((prem /\ ((z === y) <=> in(pair(x, y), f)), (in(pair(x, y), f)) <=> (prem ==> in(pair(x, y), f))) |- ((z === y) <=> (prem ==> in(pair(x, y), f)))) by RightSubstIff + .withParametersSimple( + List(((in(pair(x, y), f)), (prem ==> in(pair(x, y), f)))), + lambda(h, ((z === y) <=> h)) + ) have((prem /\ ((z === y) <=> in(pair(x, y), f)), prem) |- ((z === y) <=> (prem ==> in(pair(x, y), f)))) by Cut(iff, subst) thenHave(thesis) by Restate @@ -2386,7 +2399,7 @@ object SetTheory extends lisa.Main { val topIff = have(prem |- (!prem ==> (y === ∅)) <=> ⊤) by Restate val topSubst = have( (prem /\ ((z === y) <=> in(pair(x, y), f)), ((!prem ==> (y === ∅)) <=> ⊤)) |- ((z === y) <=> ((prem ==> in(pair(x, y), f)) /\ (!prem ==> (y === ∅)))) - ) by RightSubstIff(List(((!prem ==> (y === ∅)), ⊤)), lambda(h, ((z === y) <=> ((prem ==> in(pair(x, y), f)) /\ h))))(lhs) + ) by RightSubstIff.withParametersSimple(List(((!prem ==> (y === ∅)), ⊤)), lambda(h, ((z === y) <=> ((prem ==> in(pair(x, y), f)) /\ h))))(lhs) have((prem /\ ((z === y) <=> in(pair(x, y), f)), prem) |- ((z === y) <=> ((prem ==> in(pair(x, y), f)) /\ (!prem ==> (y === ∅))))) by Cut(topIff, topSubst) thenHave((prem, ((z === y) <=> in(pair(x, y), f))) |- ((z === y) <=> ((prem ==> in(pair(x, y), f)) /\ (!prem ==> (y === ∅))))) by Restate @@ -2417,7 +2430,7 @@ object SetTheory extends lisa.Main { have(!prem /\ ((z === y) <=> (y === ∅)) |- ((z === y) <=> (y === ∅))) by Restate val subst = thenHave( (!prem /\ ((z === y) <=> (y === ∅)), ((y === ∅)) <=> (!prem ==> (y === ∅))) |- ((z === y) <=> (!prem ==> (y === ∅))) - ) by RightSubstIff(List((((y === ∅)), (!prem ==> (y === ∅)))), lambda(h, ((z === y) <=> h))) + ) by RightSubstIff.withParametersSimple(List((((y === ∅)), (!prem ==> (y === ∅)))), lambda(h, ((z === y) <=> h))) have((!prem /\ ((z === y) <=> (y === ∅)), !prem) |- ((z === y) <=> (!prem ==> (y === ∅)))) by Cut(iff, subst) thenHave(thesis) by Restate @@ -2427,7 +2440,7 @@ object SetTheory extends lisa.Main { val topIff = have(!prem |- (prem ==> in(pair(x, y), f)) <=> ⊤) by Restate val topSubst = have( (!prem /\ ((z === y) <=> (y === ∅)), ((prem ==> in(pair(x, y), f)) <=> ⊤)) |- ((z === y) <=> ((prem ==> in(pair(x, y), f)) /\ (!prem ==> (y === ∅)))) - ) by RightSubstIff(List(((prem ==> in(pair(x, y), f)), ⊤)), lambda(h, ((z === y) <=> ((!prem ==> (y === ∅)) /\ h))))(lhs) + ) by RightSubstIff.withParametersSimple(List(((prem ==> in(pair(x, y), f)), ⊤)), lambda(h, ((z === y) <=> ((!prem ==> (y === ∅)) /\ h))))(lhs) have((!prem /\ ((z === y) <=> (y === ∅)), !prem) |- ((z === y) <=> ((prem ==> in(pair(x, y), f)) /\ (!prem ==> (y === ∅))))) by Cut(topIff, topSubst) thenHave((!prem, ((z === y) <=> (y === ∅))) |- ((z === y) <=> ((prem ==> in(pair(x, y), f)) /\ (!prem ==> (y === ∅))))) by Restate @@ -3216,7 +3229,7 @@ object SetTheory extends lisa.Main { inRangeImpliesPullbackExists of (z -> y), functionFromImpliesFunctional of (y -> powerSet(x)) ) - val xeqdom = thenHave((ydef, surjective(f, x, powerSet(x)), (relationDomain(f) === x)) |- ∃(z, in(z, x) /\ (app(f, z) === y))) by RightSubstEq.withParameters( + val xeqdom = thenHave((ydef, surjective(f, x, powerSet(x)), (relationDomain(f) === x)) |- ∃(z, in(z, x) /\ (app(f, z) === y))) by RightSubstEq.withParametersSimple( List((x, relationDomain(f))), lambda(x, ∃(z, in(z, x) /\ (app(f, z) === y))) ) @@ -3231,7 +3244,7 @@ object SetTheory extends lisa.Main { have(ydef |- ydef) by Hypothesis thenHave(ydef |- in(z, y) <=> (in(z, x) /\ !in(z, app(f, z)))) by InstantiateForall(z) thenHave((ydef, in(z, x), (app(f, z) === y)) |- in(z, y) <=> (in(z, x) /\ !in(z, app(f, z)))) by Weakening - thenHave((ydef, in(z, x), (app(f, z) === y)) |- in(z, app(f, z)) <=> (in(z, x) /\ !in(z, app(f, z)))) by RightSubstEq.withParameters( + thenHave((ydef, in(z, x), (app(f, z) === y)) |- in(z, app(f, z)) <=> (in(z, x) /\ !in(z, app(f, z)))) by RightSubstEq.withParametersSimple( List((y, app(f, z))), lambda(y, in(z, y) <=> (in(z, x) /\ !in(z, app(f, z)))) ) diff --git a/lisa-sets/src/main/scala/lisa/maths/settheory/SetTheory2.scala b/lisa-sets/src/main/scala/lisa/maths/settheory/SetTheory2.scala index 55690ed4..b3d69dcc 100644 --- a/lisa-sets/src/main/scala/lisa/maths/settheory/SetTheory2.scala +++ b/lisa-sets/src/main/scala/lisa/maths/settheory/SetTheory2.scala @@ -1,5 +1,9 @@ package lisa.maths.settheory +/** + * Revamp of the set theory library from scratch, since most of the current one is severely outdated. + * We can make a better presentation and organisation of theorems, better automation, uniform comments/latex tags, etc. + */ object SetTheory2 extends lisa.Main { import lisa.maths.settheory.SetTheory.* // import Comprehensions.* diff --git a/lisa-sets/src/main/scala/lisa/maths/settheory/orderings/Recursion.scala b/lisa-sets/src/main/scala/lisa/maths/settheory/orderings/Recursion.scala index 3e4d406f..4ca6df8a 100644 --- a/lisa-sets/src/main/scala/lisa/maths/settheory/orderings/Recursion.scala +++ b/lisa-sets/src/main/scala/lisa/maths/settheory/orderings/Recursion.scala @@ -1873,8 +1873,6 @@ object Recursion extends lisa.Main { val sPsi = SchematicPredicateLabel("P", 3) have(forall(y, in(y, initialSegment(p, x)) ==> existsOne(g, fun(g, y)))) by Restate - println(existsOne(g, fun(g, y))) - println(lastStep.bot) have(exists(w, forall(t, in(t, w) <=> exists(y, in(y, initialSegment(p, x)) /\ fun(t, y))))) by Tautology.from( lastStep, strictReplacementSchema of (A -> initialSegment(p, x), R -> lambda((y, g), fun(g, y))) @@ -1882,8 +1880,6 @@ object Recursion extends lisa.Main { } have(thesis) by Tautology.from(lastStep, funExists) - println("thesis") - println(lastStep.bot) } /** diff --git a/lisa-utils/src/main/scala/lisa/fol/Common.scala b/lisa-utils/src/main/scala/lisa/fol/Common.scala index 39bdc5df..356b95d8 100644 --- a/lisa-utils/src/main/scala/lisa/fol/Common.scala +++ b/lisa-utils/src/main/scala/lisa/fol/Common.scala @@ -480,6 +480,7 @@ trait Common { */ sealed trait SchematicAtomicLabel[A <: (Formula | (Seq[Term] |-> Formula)) & LisaObject[A]] extends AtomicLabel[A] with SchematicLabel[A] { this: A & LisaObject[A] => + val underlyingLabel: K.SchematicAtomicLabel override def rename(newid: Identifier): SchematicAtomicLabel[A] def freshRename(taken: Iterable[Identifier]): SchematicAtomicLabel[A] diff --git a/lisa-utils/src/main/scala/lisa/fol/Sequents.scala b/lisa-utils/src/main/scala/lisa/fol/Sequents.scala index 44eb047b..bd6bc966 100644 --- a/lisa-utils/src/main/scala/lisa/fol/Sequents.scala +++ b/lisa-utils/src/main/scala/lisa/fol/Sequents.scala @@ -10,7 +10,7 @@ import lisa.utils.K import scala.annotation.showAsInfix -trait Sequents extends Common with lisa.fol.Lambdas { +trait Sequents extends Common with lisa.fol.Lambdas with Predef { object SequentInstantiationRule extends ProofTactic given ProofTactic = SequentInstantiationRule @@ -30,7 +30,7 @@ trait Sequents extends Common with lisa.fol.Lambdas { * @param map * @return */ - def substituteWithProof(map: Map[SchematicLabel[_], _ <: LisaObject[_]]): (Sequent, Seq[K.SCProofStep]) = { + def instantiateWithProof(map: Map[SchematicLabel[_], _ <: LisaObject[_]], index: Int): (Sequent, Seq[K.SCProofStep]) = { val mTerm: Map[SchematicFunctionLabel[?] | Variable, LambdaExpression[Term, Term, ?]] = map.collect[SchematicFunctionLabel[?] | Variable, LambdaExpression[Term, Term, ?]](p => @@ -70,7 +70,27 @@ trait Sequents extends Common with lisa.fol.Lambdas { } } ) - (substituteUnsafe(map), substituteWithProofLikeKernel(mConn, mPred, mTerm)) + (substituteUnsafe(map), instantiateWithProofLikeKernel(mConn, mPred, mTerm, index)) + + } + + def instantiateForallWithProof(args: Seq[Term], index: Int): (Sequent, Seq[K.SCProofStep]) = { + if this.right.size != 1 then throw new IllegalArgumentException("Right side of sequent must be a single universally quantified formula") + this.right.head match { + case r @ Forall(x, f) => + val t = args.head + val newf = f.substitute(x := t) + val s0 = K.Hypothesis((newf |- newf).underlying, newf.underlying) + val s1 = K.LeftForall((r |- newf).underlying, index + 1, f.underlying, x.underlyingLabel, t.underlying) + val s2 = K.Cut((this.left |- newf).underlying, index, index + 2, r.underlying) + if args.tail.isEmpty then (this.left |- newf, Seq(s0, s1, s2)) + else + (this.left |- newf).instantiateForallWithProof(args.tail, index + 3) match { + case (s, p) => (s, Seq(s0, s1, s2) ++ p) + } + + case _ => throw new IllegalArgumentException("Right side of sequent must be a single universaly quantified formula") + } } @@ -83,10 +103,11 @@ trait Sequents extends Common with lisa.fol.Lambdas { * @param mTerm The substitution of function schemas * @return */ - def substituteWithProofLikeKernel( + def instantiateWithProofLikeKernel( mCon: Map[SchematicConnectorLabel[?], LambdaExpression[Formula, Formula, ?]], mPred: Map[SchematicPredicateLabel[?] | VariableFormula, LambdaExpression[Term, Formula, ?]], - mTerm: Map[SchematicFunctionLabel[?] | Variable, LambdaExpression[Term, Term, ?]] + mTerm: Map[SchematicFunctionLabel[?] | Variable, LambdaExpression[Term, Term, ?]], + index: Int ): Seq[K.SCProofStep] = { val premiseSequent = this.underlying val mConK = mCon.map((sl, le) => (sl.underlyingLabel, underlyingLFF(le))) @@ -104,7 +125,7 @@ trait Sequents extends Common with lisa.fol.Lambdas { ) val botK = lisa.utils.KernelHelpers.instantiateSchemaInSequent(premiseSequent, mConK, mPredK, mTermK) val smap = Map[SchematicLabel[?], LisaObject[?]]() ++ mCon ++ mPred ++ mTerm - Seq(K.InstSchema(botK, -1, mConK, mPredK, mTermK)) + Seq(K.InstSchema(botK, index, mConK, mPredK, mTermK)) } infix def +<<(f: Formula): Sequent = this.copy(left = this.left + f) diff --git a/lisa-utils/src/main/scala/lisa/prooflib/BasicStepTactic.scala b/lisa-utils/src/main/scala/lisa/prooflib/BasicStepTactic.scala index 320ea5a6..678cd18c 100644 --- a/lisa-utils/src/main/scala/lisa/prooflib/BasicStepTactic.scala +++ b/lisa-utils/src/main/scala/lisa/prooflib/BasicStepTactic.scala @@ -1111,26 +1111,43 @@ object BasicStepTactic { * </pre> */ object LeftSubstEq extends ProofTactic { - def withParameters(using lib: Library, proof: lib.Proof)( + + def withParametersSimple(using lib: Library, proof: lib.Proof)( equals: List[(F.Term, F.Term)], lambdaPhi: F.LambdaExpression[F.Term, F.Formula, ?] )(premise: proof.Fact)(bot: F.Sequent): proof.ProofTacticJudgement = { + withParameters(equals.map { case (a, b) => (F.lambda(Seq(), a), F.lambda(Seq(), b)) }, (lambdaPhi.bounds.asInstanceOf[Seq[F.SchematicTermLabel[?]]], lambdaPhi.body))(premise)(bot) + } + def withParameters(using lib: Library, proof: lib.Proof)( + equals: List[(F.LambdaExpression[F.Term, F.Term, ?], F.LambdaExpression[F.Term, F.Term, ?])], + lambdaPhi: (Seq[F.SchematicTermLabel[?]], F.Formula) + )(premise: proof.Fact)(bot: F.Sequent): proof.ProofTacticJudgement = { lazy val premiseSequent = proof.getSequent(premise).underlying lazy val botK = bot.underlying - lazy val equalsK = equals.map((p: (F.Term, F.Term)) => (p._1.underlying, p._2.underlying)) - - lazy val lambdaPhiK = F.underlyingLTF(lambdaPhi) - lazy val (s_es, t_es) = equalsK.unzip - lazy val phi_s = lambdaPhiK(s_es) - lazy val phi_t = lambdaPhiK(t_es) - lazy val equalities = equalsK map { case (s, t) => K.AtomicFormula(K.equality, Seq(s, t)) } + lazy val equalsK = equals.map(p => (p._1.underlyingLTT, p._2.underlyingLTT)) + lazy val lambdaPhiK = (lambdaPhi._1.map(_.underlyingLabel), lambdaPhi._2.underlying) + + val (s_es, t_es) = equalsK.unzip + val (phi_args, phi_body) = lambdaPhiK + if (phi_args.size != s_es.size) + return proof.InvalidProofTactic("The number of arguments of φ must be the same as the number of equalities.") + else if (equalsK.zip(phi_args).exists { case ((s, t), arg) => s.vars.size != arg.arity || t.vars.size != arg.arity }) + return proof.InvalidProofTactic("The arities of symbols in φ must be the same as the arities of equalities.") + + val phi_s = K.instantiateTermSchemas(phi_body, (phi_args zip s_es).toMap) + val phi_t = K.instantiateTermSchemas(phi_body, (phi_args zip t_es).toMap) + val sEqT_es = equalsK map { case (s, t) => + assert(s.vars.size == t.vars.size) + val base = K.AtomicFormula(K.equality, Seq(s.body, if (s.vars == t.vars) t.body else t(s.vars.map(K.VariableTerm)))) + (s.vars).foldLeft(base: K.Formula) { case (acc, s_arg) => K.BinderFormula(K.Forall, s_arg, acc) } + } if (!K.isSameSet(botK.right, premiseSequent.right)) proof.InvalidProofTactic("Right-hand side of the premise is not the same as the right-hand side of the conclusion.") else if ( - !K.isSameSet(botK.left + phi_s, premiseSequent.left ++ equalities + phi_t) && - !K.isSameSet(botK.left + phi_t, premiseSequent.left ++ equalities + phi_s) + !K.isSameSet(botK.left + phi_s, premiseSequent.left ++ sEqT_es + phi_t) && + !K.isSameSet(botK.left + phi_t, premiseSequent.left ++ sEqT_es + phi_s) ) proof.InvalidProofTactic("Left-hand side of the conclusion + φ(s_) is not the same as left-hand side of the premise + (s=t)_ + φ(t_) (or with s_ and t_ swapped).") else @@ -1146,22 +1163,38 @@ object BasicStepTactic { * </pre> */ object RightSubstEq extends ProofTactic { - def withParameters(using lib: Library, proof: lib.Proof)( + def withParametersSimple(using lib: Library, proof: lib.Proof)( equals: List[(F.Term, F.Term)], lambdaPhi: F.LambdaExpression[F.Term, F.Formula, ?] )(premise: proof.Fact)(bot: F.Sequent): proof.ProofTacticJudgement = { + withParameters(equals.map { case (a, b) => (F.lambda(Seq(), a), F.lambda(Seq(), b)) }, (lambdaPhi.bounds.asInstanceOf[Seq[F.SchematicTermLabel[?]]], lambdaPhi.body))(premise)(bot) + } - lazy val lambdaPhiK = F.underlyingLTF(lambdaPhi) + def withParameters(using lib: Library, proof: lib.Proof)( + equals: List[(F.LambdaExpression[F.Term, F.Term, ?], F.LambdaExpression[F.Term, F.Term, ?])], + lambdaPhi: (Seq[F.SchematicTermLabel[?]], F.Formula) + )(premise: proof.Fact)(bot: F.Sequent): proof.ProofTacticJudgement = { lazy val premiseSequent = proof.getSequent(premise).underlying lazy val botK = bot.underlying - lazy val equalsK = equals.map((p: (F.Term, F.Term)) => (p._1.underlying, p._2.underlying)) - - lazy val (s_es, t_es) = equalsK.unzip - lazy val phi_s = lambdaPhiK(s_es) - lazy val phi_t = lambdaPhiK(t_es) - lazy val equalities = equalsK map { case (s, t) => K.AtomicFormula(K.equality, Seq(s, t)) } + lazy val equalsK = equals.map(p => (p._1.underlyingLTT, p._2.underlyingLTT)) + lazy val lambdaPhiK = (lambdaPhi._1.map(_.underlyingLabel), lambdaPhi._2.underlying) + + val (s_es, t_es) = equalsK.unzip + val (phi_args, phi_body) = lambdaPhiK + if (phi_args.size != s_es.size) + return proof.InvalidProofTactic("The number of arguments of φ must be the same as the number of equalities.") + else if (equalsK.zip(phi_args).exists { case ((s, t), arg) => s.vars.size != arg.arity || t.vars.size != arg.arity }) + return proof.InvalidProofTactic("The arities of symbols in φ must be the same as the arities of equalities.") + + val phi_s = K.instantiateTermSchemas(phi_body, (phi_args zip s_es).toMap) + val phi_t = K.instantiateTermSchemas(phi_body, (phi_args zip t_es).toMap) + val sEqT_es = equalsK map { case (s, t) => + assert(s.vars.size == t.vars.size) + val base = K.AtomicFormula(K.equality, Seq(s.body, if (s.vars == t.vars) t.body else t(s.vars.map(K.VariableTerm)))) + (s.vars).foldLeft(base: K.Formula) { case (acc, s_arg) => K.BinderFormula(K.Forall, s_arg, acc) } + } - if (!K.isSameSet(botK.left, premiseSequent.left ++ equalities)) + if (!K.isSameSet(botK.left, premiseSequent.left ++ sEqT_es)) proof.InvalidProofTactic("Left-hand side of the conclusion is not the same as the left-hand side of the premise + (s=t)_.") else if ( !K.isSameSet(botK.right + phi_s, premiseSequent.right + phi_t) && @@ -1173,26 +1206,6 @@ object BasicStepTactic { } - def apply2(using lib: Library, proof: lib.Proof)(premise: proof.Fact)(bot: F.Sequent): proof.ProofTacticJudgement = { - lazy val premiseSequent = proof.getSequent(premise) - val premRight = F.AppliedConnector(F.Or, premiseSequent.right.toSeq) - val botRight = F.AppliedConnector(F.Or, bot.right.toSeq) - - val equalities = bot.left.toSeq.collect { case F.AppliedPredicate(F.equality, Seq(l, r)) => (l, r) } - val undereqs = equalities.toList.map(p => (p._1.underlying, p._2.underlying)) - val canReach = UnificationUtils.getContextFormula( - first = premRight, - second = botRight, - confinedTermRules = equalities, - takenTermVariables = equalities.flatMap(e => e._1.freeVariables ++ e._2.freeVariables).toSet - ) - - if (canReach.isEmpty) proof.InvalidProofTactic("Could not find a set of equalities to rewrite premise into conclusion successfully.") - else - val termLambda = canReach.get.toLambdaTF - withParameters(equalities.toList, termLambda)(premise)(bot) - - } } /** @@ -1203,26 +1216,43 @@ object BasicStepTactic { * </pre> */ object LeftSubstIff extends ProofTactic { - def apply(using lib: Library, proof: lib.Proof)( + def withParametersSimple(using lib: Library, proof: lib.Proof)( equals: List[(F.Formula, F.Formula)], lambdaPhi: F.LambdaExpression[F.Formula, F.Formula, ?] )(premise: proof.Fact)(bot: F.Sequent): proof.ProofTacticJudgement = { + withParameters(equals.map { case (a, b) => (F.lambda(Seq(), a), F.lambda(Seq(), b)) }, (lambdaPhi.bounds.asInstanceOf[Seq[F.SchematicAtomicLabel[?]]], lambdaPhi.body))(premise)(bot) + } + + def withParameters(using lib: Library, proof: lib.Proof)( + equals: List[(F.LambdaExpression[F.Term, F.Formula, ?], F.LambdaExpression[F.Term, F.Formula, ?])], + lambdaPhi: (Seq[F.SchematicAtomicLabel[?]], F.Formula) + )(premise: proof.Fact)(bot: F.Sequent): proof.ProofTacticJudgement = { lazy val premiseSequent = proof.getSequent(premise).underlying lazy val botK = bot.underlying - lazy val equalsK = equals.map((p: (F.Formula, F.Formula)) => (p._1.underlying, p._2.underlying)) - lazy val lambdaPhiK = F.underlyingLFF(lambdaPhi) - - lazy val (psi_es, tau_es) = equalsK.unzip - lazy val phi_psi = lambdaPhiK(psi_es) - lazy val phi_tau = lambdaPhiK(tau_es) - lazy val implications = equalsK map { case (s, t) => K.ConnectorFormula(K.Iff, Seq(s, t)) } + lazy val equalsK = equals.map(p => (p._1.underlyingLTF, p._2.underlyingLTF)) + lazy val lambdaPhiK = (lambdaPhi._1.map(_.underlyingLabel), lambdaPhi._2.underlying) + + val (psi_s, tau_s) = equalsK.unzip + val (phi_args, phi_body) = lambdaPhiK + if (phi_args.size != psi_s.size) + return proof.InvalidProofTactic("The number of arguments of φ must be the same as the number of equalities.") + else if (equalsK.zip(phi_args).exists { case ((s, t), arg) => s.vars.size != arg.arity || t.vars.size != arg.arity }) + return proof.InvalidProofTactic("The arities of symbols in φ must be the same as the arities of equalities.") + + val phi_psi = K.instantiatePredicateSchemas(phi_body, (phi_args zip psi_s).toMap) + val phi_tau = K.instantiatePredicateSchemas(phi_body, (phi_args zip tau_s).toMap) + val psiIffTau = equalsK map { case (s, t) => + assert(s.vars.size == t.vars.size) + val base = K.ConnectorFormula(K.Iff, Seq(s.body, if (s.vars == t.vars) t.body else t(s.vars.map(K.VariableTerm)))) + (s.vars).foldLeft(base: K.Formula) { case (acc, s_arg) => K.BinderFormula(K.Forall, s_arg, acc) } + } if (!K.isSameSet(botK.right, premiseSequent.right)) proof.InvalidProofTactic("Right-hand side of the premise is not the same as the right-hand side of the conclusion.") else if ( - !K.isSameSet(botK.left + phi_psi, premiseSequent.left ++ implications + phi_tau) && - !K.isSameSet(botK.left + phi_tau, premiseSequent.left ++ implications + phi_psi) + !K.isSameSet(botK.left + phi_psi, premiseSequent.left ++ psiIffTau + phi_tau) && + !K.isSameSet(botK.left + phi_tau, premiseSequent.left ++ psiIffTau + phi_psi) ) proof.InvalidProofTactic("Left-hand side of the conclusion + φ(ψ_) is not the same as left-hand side of the premise + (ψ ⇔ τ)_ + φ(τ_) (or with ψ_ and τ_ swapped).") else @@ -1238,22 +1268,39 @@ object BasicStepTactic { * </pre> */ object RightSubstIff extends ProofTactic { - def apply(using lib: Library, proof: lib.Proof)( + def withParametersSimple(using lib: Library, proof: lib.Proof)( equals: List[(F.Formula, F.Formula)], lambdaPhi: F.LambdaExpression[F.Formula, F.Formula, ?] )(premise: proof.Fact)(bot: F.Sequent): proof.ProofTacticJudgement = { + withParameters(equals.map { case (a, b) => (F.lambda(Seq(), a), F.lambda(Seq(), b)) }, (lambdaPhi.bounds.asInstanceOf[Seq[F.SchematicAtomicLabel[?]]], lambdaPhi.body))(premise)(bot) + } + + def withParameters(using lib: Library, proof: lib.Proof)( + equals: List[(F.LambdaExpression[F.Term, F.Formula, ?], F.LambdaExpression[F.Term, F.Formula, ?])], + lambdaPhi: (Seq[F.SchematicAtomicLabel[?]], F.Formula) + )(premise: proof.Fact)(bot: F.Sequent): proof.ProofTacticJudgement = { lazy val premiseSequent = proof.getSequent(premise).underlying lazy val botK = bot.underlying - lazy val equalsK = equals.map((p: (F.Formula, F.Formula)) => (p._1.underlying, p._2.underlying)) - lazy val lambdaPhiK = F.underlyingLFF(lambdaPhi) - - lazy val (psi_es, tau_es) = equalsK.unzip - lazy val phi_psi = lambdaPhiK(psi_es) - lazy val phi_tau = lambdaPhiK(tau_es) - lazy val implications = equalsK map { case (s, t) => K.ConnectorFormula(K.Iff, Seq(s, t)) } + lazy val equalsK = equals.map(p => (p._1.underlyingLTF, p._2.underlyingLTF)) + lazy val lambdaPhiK = (lambdaPhi._1.map(_.underlyingLabel), lambdaPhi._2.underlying) + + val (psi_s, tau_s) = equalsK.unzip + val (phi_args, phi_body) = lambdaPhiK + if (phi_args.size != psi_s.size) + return proof.InvalidProofTactic("The number of arguments of φ must be the same as the number of equalities.") + else if (equalsK.zip(phi_args).exists { case ((s, t), arg) => s.vars.size != arg.arity || t.vars.size != arg.arity }) + return proof.InvalidProofTactic("The arities of symbols in φ must be the same as the arities of equalities.") + + val phi_psi = K.instantiatePredicateSchemas(phi_body, (phi_args zip psi_s).toMap) + val phi_tau = K.instantiatePredicateSchemas(phi_body, (phi_args zip tau_s).toMap) + val psiIffTau = equalsK map { case (s, t) => + assert(s.vars.size == t.vars.size) + val base = K.ConnectorFormula(K.Iff, Seq(s.body, if (s.vars == t.vars) t.body else t(s.vars.map(K.VariableTerm)))) + (s.vars).foldLeft(base: K.Formula) { case (acc, s_arg) => K.BinderFormula(K.Forall, s_arg, acc) } + } - if (!K.isSameSet(botK.left, premiseSequent.left ++ implications)) { + if (!K.isSameSet(botK.left, premiseSequent.left ++ psiIffTau)) { proof.InvalidProofTactic("Left-hand side of the conclusion is not the same as the left-hand side of the premise + (ψ ⇔ τ)_.") } else if ( !K.isSameSet(botK.right + phi_psi, premiseSequent.right + phi_tau) && @@ -1261,7 +1308,7 @@ object BasicStepTactic { ) proof.InvalidProofTactic("Right-hand side of the conclusion + φ(ψ_) is not the same as right-hand side of the premise + φ(τ_) (or with ψ_ and τ_ swapped).") else - proof.ValidProofTactic(bot, Seq(K.RightSubstIff(botK, -1, equalsK, F.underlyingLFF(lambdaPhi))), Seq(premise)) + proof.ValidProofTactic(bot, Seq(K.RightSubstIff(botK, -1, equalsK, lambdaPhiK)), Seq(premise)) } } diff --git a/lisa-utils/src/main/scala/lisa/prooflib/ProofsHelpers.scala b/lisa-utils/src/main/scala/lisa/prooflib/ProofsHelpers.scala index da97d5e3..e612d0eb 100644 --- a/lisa-utils/src/main/scala/lisa/prooflib/ProofsHelpers.scala +++ b/lisa-utils/src/main/scala/lisa/prooflib/ProofsHelpers.scala @@ -121,7 +121,7 @@ trait ProofsHelpers { } extension (using proof: library.Proof)(fact: proof.Fact) { - def of(insts: F.SubstPair*): proof.InstantiatedFact = { + def of(insts: (F.SubstPair | F.Term)*): proof.InstantiatedFact = { proof.InstantiatedFact(fact, insts) } def statement: F.Sequent = proof.sequentOfFact(fact) diff --git a/lisa-utils/src/main/scala/lisa/prooflib/WithTheorems.scala b/lisa-utils/src/main/scala/lisa/prooflib/WithTheorems.scala index 98f077eb..49349ce9 100644 --- a/lisa-utils/src/main/scala/lisa/prooflib/WithTheorems.scala +++ b/lisa-utils/src/main/scala/lisa/prooflib/WithTheorems.scala @@ -43,13 +43,19 @@ trait WithTheorems { */ case class InstantiatedFact( fact: Fact, - insts: Seq[F.SubstPair] /*, - instsConn: Map[K.SchematicConnectorLabel, K.LambdaFormulaFormula], - instsPred: Map[K.SchematicAtomicLabel, K.LambdaTermFormula], - instsTerm: Map[K.SchematicTermLabel, K.LambdaTermTerm]*/ + insts: Seq[F.SubstPair | F.Term] ) { val baseFormula: F.Sequent = sequentOfFact(fact) - val result: F.Sequent = baseFormula.substitute(insts*) + val (result, proof) = { + val (terms, substPairs) = insts.partitionMap { + case t: F.Term => Left(t) + case sp: F.SubstPair => Right(sp) + } + + val (s1, p1) = if substPairs.isEmpty then (baseFormula, Seq()) else baseFormula.instantiateWithProof(substPairs.map(sp => (sp._1, sp._2)).toMap, -1) + val (s2, p2) = if terms.isEmpty then (s1, p1) else s1.instantiateForallWithProof(terms, p1.length - 1) + (s2, p1 ++ p2) + } } @@ -112,16 +118,8 @@ trait WithTheorems { } private def addInstantiatedFact(instFact: InstantiatedFact): Unit = { - val (s, i) = sequentAndIntOfFact(instFact.fact) - // newProofStep(BasicStepTactic.InstSchema(using library, this)(instFact.instsConn, instFact.instsPred, instFact.instsTerm)(i).asInstanceOf[ValidProofTactic]) - // newProofStep(BasicStepTactic.InstSchema(using library, this)(instFact.insts)(i).asInstanceOf[ValidProofTactic]) - val instMap = Map(instFact.insts.map(s => (s._1, (s._2.asInstanceOf: F.LisaObject[_])))*) - val instStep = { - val res = s.substituteWithProof(instMap) - - ValidProofTactic(res._1, res._2, Seq(instFact.fact))(using F.SequentInstantiationRule) - } - newProofStep(instStep) + val step = ValidProofTactic(instFact.result, instFact.proof, Seq(instFact.fact))(using F.SequentInstantiationRule) + newProofStep(step) instantiatedFacts = (instFact, steps.length - 1) :: instantiatedFacts } diff --git a/lisa-utils/src/main/scala/lisa/utils/Serialization.scala b/lisa-utils/src/main/scala/lisa/utils/Serialization.scala index 88e0e8db..3b934506 100644 --- a/lisa-utils/src/main/scala/lisa/utils/Serialization.scala +++ b/lisa-utils/src/main/scala/lisa/utils/Serialization.scala @@ -1,6 +1,7 @@ package lisa.utils -import lisa.utils.K.* +import lisa.kernel.proof.SequentCalculus.* +import lisa.utils.K.{LeftSubstEq => _, LeftSubstIff => _, RightSubstEq => _, RightSubstIff => _, _} import lisa.utils.ProofsShrink.* import java.io._ @@ -320,41 +321,49 @@ object Serialization { sequentToProofDOS(bot) proofDOS.writeInt(t1) proofDOS.writeShort(equals.size) - equals.foreach(t => - proofDOS.writeInt(lineOfTerm(t._1)) - proofDOS.writeInt(lineOfTerm(t._2)) + equals.foreach(ltts => + lttToProofDOS(ltts._1) + lttToProofDOS(ltts._2) ) - ltfToProofDOS(lambdaPhi) + proofDOS.writeShort(lambdaPhi._1.size) + lambdaPhi._1.foreach(stl => termLabelToDOS(stl, proofDOS)) + proofDOS.writeInt(lineOfFormula(lambdaPhi._2)) case RightSubstEq(bot, t1, equals, lambdaPhi) => proofDOS.writeByte(rightSubstEq) sequentToProofDOS(bot) proofDOS.writeInt(t1) proofDOS.writeShort(equals.size) - equals.foreach(t => - proofDOS.writeInt(lineOfTerm(t._1)) - proofDOS.writeInt(lineOfTerm(t._2)) + equals.foreach(ltts => + lttToProofDOS(ltts._1) + lttToProofDOS(ltts._2) ) - ltfToProofDOS(lambdaPhi) + proofDOS.writeShort(lambdaPhi._1.size) + lambdaPhi._1.foreach(stl => termLabelToDOS(stl, proofDOS)) + proofDOS.writeInt(lineOfFormula(lambdaPhi._2)) case LeftSubstIff(bot, t1, equals, lambdaPhi) => proofDOS.writeByte(leftSubstIff) sequentToProofDOS(bot) proofDOS.writeInt(t1) proofDOS.writeShort(equals.size) - equals.foreach(t => - proofDOS.writeInt(lineOfFormula(t._1)) - proofDOS.writeInt(lineOfFormula(t._2)) + equals.foreach(ltts => + ltfToProofDOS(ltts._1) + ltfToProofDOS(ltts._2) ) - lffToProofDOS(lambdaPhi) + proofDOS.writeShort(lambdaPhi._1.size) + lambdaPhi._1.foreach(stl => formulaLabelToDOS(stl, proofDOS)) + proofDOS.writeInt(lineOfFormula(lambdaPhi._2)) case RightSubstIff(bot, t1, equals, lambdaPhi) => proofDOS.writeByte(rightSubstIff) sequentToProofDOS(bot) proofDOS.writeInt(t1) proofDOS.writeShort(equals.size) - equals.foreach(t => - proofDOS.writeInt(lineOfFormula(t._1)) - proofDOS.writeInt(lineOfFormula(t._2)) + equals.foreach(ltts => + ltfToProofDOS(ltts._1) + ltfToProofDOS(ltts._2) ) - lffToProofDOS(lambdaPhi) + proofDOS.writeShort(lambdaPhi._1.size) + lambdaPhi._1.foreach(stl => formulaLabelToDOS(stl, proofDOS)) + proofDOS.writeInt(lineOfFormula(lambdaPhi._2)) case InstSchema(bot, t1, mCon, mPred, mTerm) => proofDOS.writeByte(instSchema) sequentToProofDOS(bot) @@ -561,29 +570,29 @@ object Serialization { LeftSubstEq( sequentFromProofDIS(), proofDIS.readInt(), - (1 to proofDIS.readShort()).map(_ => (termMap(proofDIS.readInt()), termMap(proofDIS.readInt()))).toList, - ltfFromProofDIS() + (1 to proofDIS.readShort()).map(_ => (lttFromProofDIS(), lttFromProofDIS())).toList, + ((1 to proofDIS.readShort()).map(_ => labelFromInputStream(proofDIS).asInstanceOf[SchematicTermLabel]).toList, formulaMap(proofDIS.readInt())) ) else if (psType == rightSubstEq) RightSubstEq( sequentFromProofDIS(), proofDIS.readInt(), - (1 to proofDIS.readShort()).map(_ => (termMap(proofDIS.readInt()), termMap(proofDIS.readInt()))).toList, - ltfFromProofDIS() + (1 to proofDIS.readShort()).map(_ => (lttFromProofDIS(), lttFromProofDIS())).toList, + ((1 to proofDIS.readShort()).map(_ => labelFromInputStream(proofDIS).asInstanceOf[SchematicTermLabel]).toList, formulaMap(proofDIS.readInt())) ) else if (psType == leftSubstIff) LeftSubstIff( sequentFromProofDIS(), proofDIS.readInt(), - (1 to proofDIS.readShort()).map(_ => (formulaMap(proofDIS.readInt()), formulaMap(proofDIS.readInt()))).toList, - lffFromProofDIS() + (1 to proofDIS.readShort()).map(_ => (ltfFromProofDIS(), ltfFromProofDIS())).toList, + ((1 to proofDIS.readShort()).map(_ => labelFromInputStream(proofDIS).asInstanceOf[SchematicAtomicLabel]).toList, formulaMap(proofDIS.readInt())) ) else if (psType == rightSubstIff) RightSubstIff( sequentFromProofDIS(), proofDIS.readInt(), - (1 to proofDIS.readShort()).map(_ => (formulaMap(proofDIS.readInt()), formulaMap(proofDIS.readInt()))).toList, - lffFromProofDIS() + (1 to proofDIS.readShort()).map(_ => (ltfFromProofDIS(), ltfFromProofDIS())).toList, + ((1 to proofDIS.readShort()).map(_ => labelFromInputStream(proofDIS).asInstanceOf[SchematicAtomicLabel]).toList, formulaMap(proofDIS.readInt())) ) else if (psType == instSchema) InstSchema( diff --git a/lisa-utils/src/test/scala/lisa/kernel/IncorrectProofsTests.scala b/lisa-utils/src/test/scala/lisa/kernel/IncorrectProofsTests.scala index 05933f01..8a9ba63a 100644 --- a/lisa-utils/src/test/scala/lisa/kernel/IncorrectProofsTests.scala +++ b/lisa-utils/src/test/scala/lisa/kernel/IncorrectProofsTests.scala @@ -40,23 +40,23 @@ class IncorrectProofsTests extends ProofCheckerSuite { SCProof( Hypothesis(emptySeq +<< (x === y) +>> (x === y), x === y), - RightSubstEq(emptySeq +<< (x === y) +<< (x === z) +>> (z === y), 0, List((x, z)), LambdaTermFormula(Seq(yl), x === y)) // wrong variable replaced + RightSubstEq(emptySeq +<< (x === y) +<< (x === z) +>> (z === y), 0, List(((LambdaTermTerm(Seq(), x), LambdaTermTerm(Seq(), z)))), (Seq(yl), x === y)) // wrong variable replaced ), SCProof( Hypothesis(emptySeq +<< (x === y) +>> (x === y), x === y), - RightSubstEq(emptySeq +<< (x === y) +>> (z === y), 0, List((x, z)), LambdaTermFormula(Seq(xl), x === y)) // missing hypothesis + RightSubstEq(emptySeq +<< (x === y) +>> (z === y), 0, List(((LambdaTermTerm(Seq(), x), LambdaTermTerm(Seq(), z)))), (Seq(xl), x === y)) // missing hypothesis ), SCProof( Hypothesis(emptySeq +<< (x === y) +>> (x === y), x === y), - RightSubstEq(emptySeq +<< (x === y) +<< (x === z) +>> (z === y), 0, List((x, z)), LambdaTermFormula(Seq(xl), x === z)) // replacement mismatch + RightSubstEq(emptySeq +<< (x === y) +<< (x === z) +>> (z === y), 0, List(((LambdaTermTerm(Seq(), x), LambdaTermTerm(Seq(), z)))), (Seq(xl), x === z)) // replacement mismatch ), SCProof( Hypothesis(emptySeq +<< (x === y) +>> (x === y), x === y), - LeftSubstEq(emptySeq +<< (z === y) +<< (x === z) +>> (x === y), 0, List((x, z)), LambdaTermFormula(Seq(yl), x === y)) + LeftSubstEq(emptySeq +<< (z === y) +<< (x === z) +>> (x === y), 0, List(((LambdaTermTerm(Seq(), x), LambdaTermTerm(Seq(), z)))), (Seq(yl), x === y)) ), SCProof( Hypothesis(emptySeq +<< (f <=> g) +>> (f <=> g), f <=> g), - LeftSubstIff(emptySeq +<< (h <=> g) +<< (f <=> h) +>> (f <=> g), 0, List((f, h)), LambdaFormulaFormula(Seq(gl), f <=> g)) + LeftSubstIff(emptySeq +<< (h <=> g) +<< (f <=> h) +>> (f <=> g), 0, List(((LambdaTermFormula(Seq(), f), LambdaTermFormula(Seq(), h)))), (Seq(gl), f <=> g)) ), SCProof( Hypothesis(emptySeq +<< f +>> f, f), diff --git a/lisa-utils/src/test/scala/lisa/kernel/ProofTests.scala b/lisa-utils/src/test/scala/lisa/kernel/ProofTests.scala index 474f4fee..f86a1813 100644 --- a/lisa-utils/src/test/scala/lisa/kernel/ProofTests.scala +++ b/lisa-utils/src/test/scala/lisa/kernel/ProofTests.scala @@ -5,6 +5,7 @@ import lisa.kernel.proof.RunningTheory import lisa.kernel.proof.RunningTheory.* import lisa.kernel.proof.SCProof import lisa.kernel.proof.SCProofChecker +import lisa.kernel.proof.SCProofChecker.checkSCProof import lisa.kernel.proof.SequentCalculus.* import lisa.utils.KernelHelpers.{_, given} import lisa.utils.Printer @@ -14,16 +15,25 @@ import scala.language.adhocExtensions import scala.util.Random class ProofTests extends AnyFunSuite { - val predicateVerifier = SCProofChecker.checkSCProof private val x = VariableLabel("x") private val y = VariableLabel("y") private val z = VariableLabel("z") + val f = SchematicFunctionLabel("f", 1) + val f2 = SchematicFunctionLabel("f2", 1) + val g = ConstantFunctionLabel("g", 2) + val g2 = SchematicFunctionLabel("g2", 2) private val a = AtomicFormula(ConstantAtomicLabel("A", 0), Seq()) private val b = AtomicFormula(ConstantAtomicLabel("B", 0), Seq()) private val fp = ConstantAtomicLabel("F", 1) val sT = VariableLabel("t") + val X = VariableFormulaLabel("X") + val P = SchematicPredicateLabel("f", 1) + val P2 = SchematicPredicateLabel("f2", 1) + val Q = ConstantAtomicLabel("g", 2) + val Q2 = SchematicPredicateLabel("g2", 2) + test("Verification of Pierce law") { val s0 = Hypothesis(a |- a, a) val s1 = Weakening(a |- Set(a, b), 0) @@ -31,14 +41,203 @@ class ProofTests extends AnyFunSuite { val s3 = LeftImplies((a ==> b) ==> a |- a, 2, 0, a ==> b, a) val s4 = RightImplies(() |- (a ==> b) ==> a ==> a, 3, (a ==> b) ==> a, a) val ppl: SCProof = SCProof(IndexedSeq(s0, s1, s2, s3, s4)) - assert(predicateVerifier(ppl).isValid) + assert(checkSCProof(ppl).isValid) } - test("Verification of substitution") { - val t0 = Hypothesis(fp(x) |- fp(x), fp(x)) - val t1 = RightSubstEq(Set(fp(x), x === y) |- fp(y), 0, List((x, y)), LambdaTermFormula(Seq(sT), fp(sT))) - val pr = new SCProof(IndexedSeq(t0, t1)) - assert(predicateVerifier(pr).isValid) + test("Verification of LeftSubstEq") { + { + val t0 = Hypothesis(fp(x) |- fp(x), fp(x)) + val t1 = LeftSubstEq(Set(fp(y), x === y) |- fp(x), 0, List(((LambdaTermTerm(Seq(), x), LambdaTermTerm(Seq(), y)))), (Seq(sT), fp(sT))) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, fp(f(x))) |- exists(x, fp(f(x))), exists(x, fp(f(x)))) + val t1 = LeftSubstEq( + Set(exists(x, fp(g(x, x))), forall(y, f(y) === g(y, y))) |- exists(x, fp(f(x))), + 0, + List((LambdaTermTerm(Seq(x), f(x)), LambdaTermTerm(Seq(x), g(x, x)))), + (Seq(f2), exists(x, fp(f2(x)))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, fp(f(x))) |- exists(x, fp(f(x))), exists(x, fp(f(x)))) + val t1 = LeftSubstEq( + Set(exists(x, fp(g(x, x))), forall(y, f(y) === g(y, y))) |- exists(x, fp(f(x))), + 0, + List((LambdaTermTerm(Seq(y), f(y)), LambdaTermTerm(Seq(z), g(z, z)))), + (Seq(f2), exists(x, fp(f2(x)))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, forall(y, fp(g(y, g(x, z))))) |- exists(x, forall(y, fp(g(y, g(x, z))))), exists(x, forall(y, fp(g(y, g(x, z)))))) + val t1 = LeftSubstEq( + Set(exists(x, forall(y, fp(g(g(x, z), y)))), forall(y, forall(z, g(y, z) === g(z, y)))) |- exists(x, forall(y, fp(g(y, g(x, z))))), + 0, + List((LambdaTermTerm(Seq(y, z), g(y, z)), LambdaTermTerm(Seq(y, z), g(z, y)))), + (Seq(g2), exists(x, forall(y, fp(g2(y, g(x, z)))))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, forall(y, fp(g(y, g(x, z))))) |- exists(x, forall(y, fp(g(y, g(x, z))))), exists(x, forall(y, fp(g(y, g(x, z)))))) + val t1 = LeftSubstEq( + Set(exists(x, forall(y, fp(g(g(z, x), y)))), forall(y, forall(z, g(y, z) === g(z, y)))) |- exists(x, forall(y, fp(g(y, g(x, z))))), + 0, + List((LambdaTermTerm(Seq(y, z), g(y, z)), LambdaTermTerm(Seq(y, z), g(z, y)))), + (Seq(g2), exists(x, forall(y, fp(g2(y, g2(x, z)))))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, forall(y, fp(g(y, g(x, f(z)))))) |- exists(x, forall(y, fp(g(y, g(x, f(z)))))), exists(x, forall(y, fp(g(y, g(x, f(z))))))) + val t1 = LeftSubstEq( + Set(exists(x, forall(y, fp(g(g(g(z, z), x), y)))), forall(y, f(y) === g(y, y)), forall(y, forall(z, g(y, z) === g(z, y)))) |- exists(x, forall(y, fp(g(y, g(x, f(z)))))), + 0, + List((LambdaTermTerm(Seq(y, z), g(y, z)), LambdaTermTerm(Seq(y, z), g(z, y))), (LambdaTermTerm(Seq(y), f(y)), LambdaTermTerm(Seq(z), g(z, z)))), + (Seq(g2, f2), exists(x, forall(y, fp(g2(y, g2(x, f2(z))))))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + } + + test("Verification of RightSubstEq") { + { + val t0 = Hypothesis(fp(x) |- fp(x), fp(x)) + val t1 = RightSubstEq(Set(fp(x), x === y) |- fp(y), 0, List(((LambdaTermTerm(Seq(), x), LambdaTermTerm(Seq(), y)))), (Seq(sT), fp(sT))) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, fp(f(x))) |- exists(x, fp(f(x))), exists(x, fp(f(x)))) + val t1 = RightSubstEq( + Set(exists(x, fp(f(x))), forall(y, f(y) === g(y, y))) |- exists(x, fp(g(x, x))), + 0, + List((LambdaTermTerm(Seq(x), f(x)), LambdaTermTerm(Seq(x), g(x, x)))), + (Seq(f2), exists(x, fp(f2(x)))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, fp(f(x))) |- exists(x, fp(f(x))), exists(x, fp(f(x)))) + val t1 = RightSubstEq( + Set(exists(x, fp(f(x))), forall(y, f(y) === g(y, y))) |- exists(x, fp(g(x, x))), + 0, + List((LambdaTermTerm(Seq(y), f(y)), LambdaTermTerm(Seq(z), g(z, z)))), + (Seq(f2), exists(x, fp(f2(x)))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, forall(y, fp(g(y, g(x, z))))) |- exists(x, forall(y, fp(g(y, g(x, z))))), exists(x, forall(y, fp(g(y, g(x, z)))))) + val t1 = RightSubstEq( + Set(exists(x, forall(y, fp(g(y, g(x, z))))), forall(y, forall(z, g(y, z) === g(z, y)))) |- exists(x, forall(y, fp(g(g(x, z), y)))), + 0, + List((LambdaTermTerm(Seq(y, z), g(y, z)), LambdaTermTerm(Seq(y, z), g(z, y)))), + (Seq(g2), exists(x, forall(y, fp(g2(y, g(x, z)))))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, forall(y, fp(g(y, g(x, z))))) |- exists(x, forall(y, fp(g(y, g(x, z))))), exists(x, forall(y, fp(g(y, g(x, z)))))) + val t1 = RightSubstEq( + Set(exists(x, forall(y, fp(g(y, g(x, z))))), forall(y, forall(z, g(y, z) === g(z, y)))) |- exists(x, forall(y, fp(g(g(z, x), y)))), + 0, + List((LambdaTermTerm(Seq(y, z), g(y, z)), LambdaTermTerm(Seq(y, z), g(z, y)))), + (Seq(g2), exists(x, forall(y, fp(g2(y, g2(x, z)))))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, forall(y, fp(g(y, g(x, f(z)))))) |- exists(x, forall(y, fp(g(y, g(x, f(z)))))), exists(x, forall(y, fp(g(y, g(x, f(z))))))) + val t1 = RightSubstEq( + Set(exists(x, forall(y, fp(g(y, g(x, f(z)))))), forall(y, f(y) === g(y, y)), forall(y, forall(z, g(y, z) === g(z, y)))) |- exists(x, forall(y, fp(g(g(g(z, z), x), y)))), + 0, + List((LambdaTermTerm(Seq(y, z), g(y, z)), LambdaTermTerm(Seq(y, z), g(z, y))), (LambdaTermTerm(Seq(y), f(y)), LambdaTermTerm(Seq(z), g(z, z)))), + (Seq(g2, f2), exists(x, forall(y, fp(g2(y, g2(x, f2(z))))))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + + } + + test("Verification of LeftSubstIff") { + { + val t0 = Hypothesis(P(x) |- P(x), P(x)) + val t1 = LeftSubstIff(Set(P(y), P(x) <=> P(y)) |- P(x), 0, List(((LambdaTermFormula(Seq(), P(x)), LambdaTermFormula(Seq(), P(y))))), (Seq(X), X)) + val pr = new SCProof(IndexedSeq(t0, t1)) + assert(checkSCProof(pr).isValid) + } + { + val t0 = Hypothesis(exists(x, P(x)) |- exists(x, P(x)), exists(x, P(x))) + val t1 = LeftSubstIff( + Set(exists(x, Q(x, x)), forall(y, P(y) <=> Q(y, y))) |- exists(x, P(x)), + 0, + List((LambdaTermFormula(Seq(x), P(x)), LambdaTermFormula(Seq(x), Q(x, x)))), + (Seq(P2), exists(x, P2(x))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, P(x)) |- exists(x, P(x)), exists(x, P(x))) + val t1 = LeftSubstIff( + Set(exists(x, Q(x, x)), forall(y, P(y) <=> Q(y, y))) |- exists(x, P(x)), + 0, + List((LambdaTermFormula(Seq(y), P(y)), LambdaTermFormula(Seq(z), Q(z, z)))), + (Seq(P2), exists(x, P2(x))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, forall(y, Q(y, g(x, z)))) |- exists(x, forall(y, Q(y, g(x, z)))), exists(x, forall(y, Q(y, g(x, z))))) + val t1 = LeftSubstIff( + Set(exists(x, forall(y, Q(g(x, z), y))), forall(x, forall(y, Q(x, y) <=> Q(y, x)))) |- exists(x, forall(y, Q(y, g(x, z)))), + 0, + List((LambdaTermFormula(Seq(y, z), Q(y, z)), LambdaTermFormula(Seq(y, z), Q(z, y)))), + (Seq(Q2), exists(x, forall(y, Q2(y, g(x, z))))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + + } + + test("Verification of RightSubstIff") { + { + val t0 = Hypothesis(P(x) |- P(x), P(x)) + val t1 = RightSubstIff(Set(P(x), P(x) <=> P(y)) |- P(y), 0, List(((LambdaTermFormula(Seq(), P(x)), LambdaTermFormula(Seq(), P(y))))), (Seq(X), X)) + val pr = new SCProof(IndexedSeq(t0, t1)) + assert(checkSCProof(pr).isValid) + } + { + val t0 = Hypothesis(exists(x, P(x)) |- exists(x, P(x)), exists(x, P(x))) + val t1 = RightSubstIff( + Set(exists(x, P(x)), forall(y, P(y) <=> Q(y, y))) |- exists(x, Q(x, x)), + 0, + List((LambdaTermFormula(Seq(x), P(x)), LambdaTermFormula(Seq(x), Q(x, x)))), + (Seq(P2), exists(x, P2(x))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, P(x)) |- exists(x, P(x)), exists(x, P(x))) + val t1 = RightSubstIff( + Set(exists(x, P(x)), forall(y, P(y) <=> Q(y, y))) |- exists(x, Q(x, x)), + 0, + List((LambdaTermFormula(Seq(y), P(y)), LambdaTermFormula(Seq(z), Q(z, z)))), + (Seq(P2), exists(x, P2(x))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } + { + val t0 = Hypothesis(exists(x, forall(y, Q(y, g(x, z)))) |- exists(x, forall(y, Q(y, g(x, z)))), exists(x, forall(y, Q(y, g(x, z))))) + val t1 = RightSubstIff( + Set(exists(x, forall(y, Q(y, g(x, z)))), forall(x, forall(y, Q(x, y) <=> Q(y, x)))) |- exists(x, forall(y, Q(g(x, z), y))), + 0, + List((LambdaTermFormula(Seq(y, z), Q(y, z)), LambdaTermFormula(Seq(y, z), Q(z, y)))), + (Seq(Q2), exists(x, forall(y, Q2(g(x, z), y)))) + ) + assert(checkSCProof(SCProof(IndexedSeq(t0, t1))).isValid) + } } test("Commutativity on a random large formula") { @@ -58,6 +257,6 @@ class ProofTests extends AnyFunSuite { val orig = subformulas.next().head val swapped = subformulasSwapped.next().head val prf = SCProof(Vector(Hypothesis(Sequent(Set(orig), Set(orig)), orig), Restate(Sequent(Set(orig), Set(swapped)), 0))) - assert(predicateVerifier(prf).isValid) + assert(checkSCProof(prf).isValid) } } diff --git a/refman/kernel.tex b/refman/kernel.tex index 5d5aeb19..340d278c 100644 --- a/refman/kernel.tex +++ b/refman/kernel.tex @@ -507,26 +507,39 @@ Finally, the two rules Restate and Weakening leverage the $\FOLalg$ algorithm. \RightLabel{\text{ InstSchema}} \UnaryInfC{$\Gamma[\psi(\vec{v}) := {\schem{p}(\vec{v})}] \vdash \Delta[\psi(\vec{v}) := {\schem{p}(\vec{v})}]$} \DisplayProof - } \\[5ex] + } + \\[5ex] - \AxiomC{$\Gamma, \phi[\schem{f} := s] \vdash \Delta$} - \RightLabel{\text{ LeftSubstEq}} - \UnaryInfC{$\Gamma, s = t, \phi[\schem{f} := t] \vdash \Delta$} - \DisplayProof & - \AxiomC{$\Gamma \vdash \phi[\schem{f} := s], \Delta$} - \RightLabel{\text{ RightSubstEq}} - \UnaryInfC{$\Gamma, s = t \vdash \phi[\schem{f} := t], \Delta$} - \DisplayProof + \multicolumn{2}{c}{ + \AxiomC{$\Gamma, \phi[\schem{f} := s] \vdash \Delta$} + \RightLabel{\text{ LeftSubstEq}} + \UnaryInfC{$\Gamma, \forall \vec x. s(\vec x) = t(\vec x), \phi[\schem{f} := t] \vdash \Delta$} + \DisplayProof + } \\[5ex] - \AxiomC{$\Gamma, \phi[{\schem{p}} := a] \vdash \Delta$} - \RightLabel{\text{ LeftSubstIff}} - \UnaryInfC{$\Gamma, a \leftrightarrow b, \phi[{\schem{p}} = b] \vdash \Delta$} - \DisplayProof & - \AxiomC{$\Gamma \vdash \phi[{\schem{p}} := a], \Delta$} - \RightLabel{\text{ RightSubstIff}} - \UnaryInfC{$\Gamma, a \leftrightarrow b \vdash \phi[{\schem{p}} := b], \Delta$} - \DisplayProof + \multicolumn{2}{c}{ + \AxiomC{$\Gamma \vdash \phi[\schem{f} := s], \Delta$} + \RightLabel{\text{ RightSubstEq}} + \UnaryInfC{$\Gamma, \forall \vec x. s(\vec x) = t(\vec x) \vdash \phi[\schem{f} := t], \Delta$} + \DisplayProof + } + \\[5ex] + + \multicolumn{2}{c}{ + \AxiomC{$\Gamma, \phi[{\schem{p}} := a] \vdash \Delta$} + \RightLabel{\text{ LeftSubstIff}} + \UnaryInfC{$\Gamma, \forall \vec x. a(\vec x) \leftrightarrow b(\vec x), \phi[{\schem{p}} = b] \vdash \Delta$} + \DisplayProof + } + \\[5ex] + + \multicolumn{2}{c}{ + \AxiomC{$\Gamma \vdash \phi[{\schem{p}} := a], \Delta$} + \RightLabel{\text{ RightSubstIff}} + \UnaryInfC{$\Gamma, \forall \vec x. a(\vec x) \leftrightarrow b(\vec x) \vdash \phi[{\schem{p}} := b], \Delta$} + \DisplayProof + } \\[5ex] \AxiomC{$\Gamma, t = t \vdash \Delta$} @@ -562,7 +575,7 @@ Finally, the two rules Restate and Weakening leverage the $\FOLalg$ algorithm. \multicolumn{2}{c}{ \AxiomC{} - \RightLabel{\text { Sorry: admit a statement as axiom. Usage transitively tracked.}} + \RightLabel{\text { Sorry: admit a statement without proof. 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zF<<A$V%3F@M34SDdCB1ZBg1L!o!i&k@efmp42s_~X=dFftviznK87v-<(auumG_AD z>aUYa=j&vu-fQDoE!`D<_2jE<2Y0;-7JT&feokq7!e8~(%15n(vP<VwxJyUe_1M!s zf9*S+dyDUdxp#iwx^8*&hPbTK<ob_wyYfn#D(d^T?CPxezrs3v?~&in9ahiwQmC1K z^>6dn`)m8AIV|Q{9o+mie{G&x<BO8j?5wXJhaLA=;C}b3tkjzSqBh2jEjr#t8+WoS z`&sUqaO?cE{)q=>sfKNza9~$lm^e#z{XW5n?~7GW-Vkqp>~ZE;j7+<*fxNo+0<}{+ y_5U;5`mbZ>O3h0tE-6Y)%muOXld`yqOA?DpDvDCmxQvX=EDTM#R8?L5-M9d#2>+)5 diff --git a/refman/macro.tex b/refman/macro.tex index 2919e649..c3f28633 100644 --- a/refman/macro.tex +++ b/refman/macro.tex @@ -96,6 +96,7 @@ \definecolor{comments}{RGB}{80,0,110} \lstdefinelanguage{scala}{ + keepspaces=true, extendedchars = true, inputencoding = utf8, alsoletter={@,=,>}, @@ -119,7 +120,9 @@ otherkeywords = {;,<<,>>,++}, } + \lstdefinelanguage{lisa}{ + keepspaces=true, extendedchars = true, inputencoding = utf8, alsoletter={@,=,>}, @@ -151,6 +154,7 @@ \definecolor{bluishlightgray}{rgb}{0.94, 0.96, 0.98} \lstdefinelanguage{console}{ + keepspaces=true, extendedchars = true, inputencoding = utf8, backgroundcolor = \color{bluishlightgray}, @@ -164,6 +168,7 @@ } \lstset{ + keepspaces=true, extendedchars = true, inputencoding = utf8, extendedchars = \true, diff --git a/refman/prooflib.tex b/refman/prooflib.tex index b338295a..08853d82 100644 --- a/refman/prooflib.tex +++ b/refman/prooflib.tex @@ -93,10 +93,38 @@ This simulation is provided by Lisa through the \lstinline|witness|{} method. It \section{DSL} -%It is important to note that when multiple such files are developed, they all use the same underlying \lstinline|RunningTheory|{}. This makes it possible to use results proved previously by means of a simple \lstinline|import|{} statement as one would import a regular object. Similarly, one should also import as usual automation and tactics developed alongside. It is expected in the medium term that \lstinline|lisa.Main|{} will come with basic automation. -%All imported theory objects will be run through as well, but only the result of the selected one will be printed. +\subsection{Instantiations with ``of''} +With lisa's kernel, it is possible to instantiate a theorem proving $P(x)$ to obtain a proof of $P(t)$, for any term $t$, using the \texttt{Inst} rule from \autoref{fig:deduct_rules_1}. Lisa's DSL provides a more convenient way to do so, using the \lstinline|of| keyword. It is used like so: +\begin{lstlisting}[language=lisa, frame=single] + val ax = Axiom(P(x)) + val falso = Theorem(P(c) /\ P(d)) { + have(thesis) by RightAnd(ax of (x := c), ax of (x := d)) + } +\end{lstlisting} +\lstinline|x := d| is called a substitution pair, and is equivalent to the tuple \lstinline|(x, d)|. Arbitrarily many substitution pairs can be given as argument to \lstinline|of|, and the instantiations are done simultaneously. \lstinline|ax of (x := c)| is called an \lstinline|InstantiatedFact|, whose statement is $P(c)$, and which can be used exactly like theorems, axioms and intermediate steps in the proof. Internally, Lisa produces a proof step corresponding to the instantiation using the \texttt{Inst} rule. +The \lstinline|of| keyword can also instantiate universaly quantified formulas of a fact, when it contains a single formula. For example, the following code is valid: +\begin{lstlisting}[language=lisa, frame=single] + val ax = Axiom(∀(x, P(x))) + val thm = Theorem(P(c) /\ P(d)) { + have(thesis) by RightAnd(ax of c, ax of d) + } +\end{lstlisting} +Here, \lstinline|ax of c| is a fact whose proven statement is again $P(c)$. It is possible to instantiate multiple $\forall$ quantifiers at once. For example if \lstinline|ax| is an axiom of the form $\forall x, \forall y, P(x, y)$, then \lstinline|ax of (c, d)| is a fact whose proven statement is $P(c, d)$. It is also possible to combine instantiation of free symbols and quantified variables. For example, if \lstinline|ax| is an axiom of the form $\forall x, \forall y, P(x, y)$, then \lstinline|ax of (c, y, P := ===)| is a fact whose proven statement is $(c = y)$. +Formally, the \lstinline|of| keyword takes as argument arbitrarily many terms and substitution pairs. If there is at least on term given as argument, the base fact must have a single universaly quantified formula on the right (an arbitrarily many formulas on the left). The number of given terms must be at most the number of leading universal quantifiers. Moreover, a substitution cannot instantiate any locked symbol (i.e. a symbol part of an assumption or definition). The ordering of substitution pairs does not matter, but the ordering of terms does. The resulting fact is obtained by firstreplacing the free symbols in the formula by the given substitution pairs, and then instantiating the quantified variables in the formula by the given terms +In general, for the following proof +\begin{lstlisting}[language=lisa, frame=single] + val ax = Axiom(∀(x, ∀(y, P(x, y)))) + val thm = Theorem(c == d) { + have(thesis) by Restate.from(ax of (c, d, P := ===)) + } +\end{lstlisting} +Lisa will produce the following inner statements: +\begin{lstlisting} + -1 Import 0 ( ) ⊢ ∀(x, ∀(y, P(x, y))) + 0 SequentInstantiationRule ( ) ⊢ c === d +\end{lstlisting} -- GitLab