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Commit 73a298ee authored by Katja Goltsova's avatar Katja Goltsova Committed by Viktor Kunčak
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Factor out Peano induction application

parent 152289c4
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3 merge requests!54Front integration,!53Front integration,!52Front integration
This commit is part of merge request !53. Comments created here will be created in the context of that merge request.
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src/main/scala/lisa/proven/mathematics/Peano.scala
+ 58
− 74
View file @ 73a298ee
...@@ -6,8 +6,9 @@ import lisa.kernel.proof.SCProof ...@@ -6,8 +6,9 @@ import lisa.kernel.proof.SCProof
import lisa.kernel.proof.SequentCalculus.* import lisa.kernel.proof.SequentCalculus.*
import lisa.proven.PeanoArithmeticsLibrary import lisa.proven.PeanoArithmeticsLibrary
import lisa.proven.tactics.ProofTactics.* import lisa.proven.tactics.ProofTactics.*
import lisa.utils.Helpers.{_, given} import lisa.utils.Helpers.{*, given}
import lisa.utils.Library import lisa.utils.Library
import lisa.utils.Printer
object Peano { object Peano {
export lisa.proven.PeanoArithmeticsLibrary.{*, given} export lisa.proven.PeanoArithmeticsLibrary.{*, given}
...@@ -32,6 +33,36 @@ object Peano { ...@@ -32,6 +33,36 @@ object Peano {
Proof(IndexedSeq(p1, p2, p3), IndexedSeq(ax)) Proof(IndexedSeq(p1, p2, p3), IndexedSeq(ax))
} }
def applyInduction(baseProof: SCSubproof, inductionStepProof: SCSubproof, inductionInstance: SCProofStep): IndexedSeq[SCProofStep] = {
require(baseProof.bot.right.size == 1, s"baseProof should prove exactly one formula, got ${Printer.prettySequent(baseProof.bot)}")
require(inductionStepProof.bot.right.size == 1, s"inductionStepProof should prove exactly one formula, got ${Printer.prettySequent(inductionStepProof.bot)}")
require(
inductionInstance.bot.left.isEmpty && inductionInstance.bot.right.size == 1,
s"induction instance step should have nothing on the left and exactly one formula on the right, got ${Printer.prettySequent(inductionInstance.bot)}"
)
val (premise, conclusion) = (inductionInstance.bot.right.head match {
case ConnectorFormula(Implies, Seq(premise, conclusion)) => (premise, conclusion)
case _ => require(false, s"induction instance should be of the form A => B, got ${Printer.prettyFormula(inductionInstance.bot.right.head)}")
})
val baseFormula = baseProof.bot.right.head
val stepFormula = inductionStepProof.bot.right.head
require(
isSame(baseFormula /\ stepFormula, premise),
"induction instance premise should be of the form base /\\ step, got " +
s"premise: ${Printer.prettyFormula(premise)}, base: ${Printer.prettyFormula(baseFormula)}, step: ${Printer.prettyFormula(stepFormula)}"
)
val lhs: Set[Formula] = baseProof.bot.left ++ inductionStepProof.bot.left
val base0 = baseProof
val step1 = inductionStepProof
val instance2 = inductionInstance
val inductionPremise3 = RightAnd(lhs |- baseFormula /\ stepFormula, Seq(0, 1), Seq(baseFormula, stepFormula))
val hypConclusion4 = hypothesis(conclusion)
val inductionInstanceOnTheLeft5 = LeftImplies(lhs + (premise ==> conclusion) |- conclusion, 3, 4, premise, conclusion)
val cutInductionInstance6 = Cut(lhs |- conclusion, 2, 5, premise ==> conclusion)
IndexedSeq(base0, step1, instance2, inductionPremise3, hypConclusion4, inductionInstanceOnTheLeft5, cutInductionInstance6)
}
THEOREM("x + 0 = 0 + x") of "∀x. plus(x, zero) === plus(zero, x)" PROOF { THEOREM("x + 0 = 0 + x") of "∀x. plus(x, zero) === plus(zero, x)" PROOF {
val refl0: SCProofStep = RightRefl(() |- s(x) === s(x), s(x) === s(x)) val refl0: SCProofStep = RightRefl(() |- s(x) === s(x), s(x) === s(x))
val subst1 = RightSubstEq((x === plus(zero, x)) |- s(x) === s(plus(zero, x)), 0, (x, plus(zero, x)) :: Nil, LambdaTermFormula(Seq(y), s(x) === s(y))) val subst1 = RightSubstEq((x === plus(zero, x)) |- s(x) === s(plus(zero, x)), 0, (x, plus(zero, x)) :: Nil, LambdaTermFormula(Seq(y), s(x) === s(y)))
...@@ -87,18 +118,7 @@ object Peano { ...@@ -87,18 +118,7 @@ object Peano {
x x
) )
val hypConclusion13 = hypothesis(forall(x, plus(x, zero) === plus(zero, x))) val inductionInstance: SCProofStep = InstPredSchema(
val inductionLhs14 = LeftImplies(
forall(x, (plus(x, zero) === plus(zero, x)) ==> (plus(s(x), zero) === plus(zero, s(x)))) ==> forall(x, plus(x, zero) === plus(zero, x)) |- forall(
x,
plus(x, zero) === plus(zero, x)
),
12,
13,
forall(x, (plus(x, zero) === plus(zero, x)) ==> (plus(s(x), zero) === plus(zero, s(x)))),
forall(x, plus(x, zero) === plus(zero, x))
)
val inductionInstance15: SCProofStep = InstPredSchema(
() |- ((plus(zero, zero) === plus(zero, zero)) /\ forall(x, (plus(x, zero) === plus(zero, x)) ==> (plus(s(x), zero) === plus(zero, s(x))))) ==> forall( () |- ((plus(zero, zero) === plus(zero, zero)) /\ forall(x, (plus(x, zero) === plus(zero, x)) ==> (plus(s(x), zero) === plus(zero, s(x))))) ==> forall(
x, x,
plus(x, zero) === plus(zero, x) plus(x, zero) === plus(zero, x)
...@@ -106,35 +126,18 @@ object Peano { ...@@ -106,35 +126,18 @@ object Peano {
-3, -3,
Map(sPhi -> LambdaTermFormula(Seq(x), plus(x, zero) === plus(zero, x))) Map(sPhi -> LambdaTermFormula(Seq(x), plus(x, zero) === plus(zero, x)))
) )
val commutesWithZero16 = Cut(
() |- forall(x, plus(x, zero) === plus(zero, x)),
15,
14,
((plus(zero, zero) === plus(zero, zero)) /\ forall(x, (plus(x, zero) === plus(zero, x)) ==> (plus(s(x), zero) === plus(zero, s(x))))) ==> forall(
x,
plus(x, zero) === plus(zero, x)
)
)
SCProof( SCProof(
IndexedSeq( applyInduction(
refl0, SCSubproof(SCProof(RightRefl(() |- zero === zero, zero === zero))),
subst1, SCSubproof(
implies2, SCProof(
transform3, IndexedSeq(refl0, subst1, implies2, transform3, instanceAx4_4, cut5, transform6, leftAnd7, instancePlusZero8, instancePlusZero9, rightAnd10, cut11, forall12),
instanceAx4_4, IndexedSeq(ax"ax4plusSuccessor", ax"ax3neutral")
cut5, ),
transform6, Seq(-1, -2)
leftAnd7, ),
instancePlusZero8, inductionInstance
instancePlusZero9,
rightAnd10,
cut11,
forall12,
hypConclusion13,
inductionLhs14,
inductionInstance15,
commutesWithZero16
), ),
IndexedSeq(ax"ax4plusSuccessor", ax"ax3neutral", ax"ax7induction") IndexedSeq(ax"ax4plusSuccessor", ax"ax3neutral", ax"ax7induction")
) )
...@@ -231,41 +234,22 @@ object Peano { ...@@ -231,41 +234,22 @@ object Peano {
) )
} }
val inductionOnY2 = Rewrite(() |- (sPhi(zero) /\ forall(y, sPhi(y) ==> sPhi(s(y)))) ==> forall(y, sPhi(y)), -3) val inductionInstance = {
val inductionInstance3 = InstPredSchema( val inductionOnY0 = Rewrite(() |- (sPhi(zero) /\ forall(y, sPhi(y) ==> sPhi(s(y)))) ==> forall(y, sPhi(y)), -1)
() |- val inductionInstance1 = InstPredSchema(
((plus(s(x), zero) === plus(x, s(zero))) /\ () |-
forall(y, (plus(x, s(y)) === plus(s(x), y)) ==> (plus(x, s(s(y))) === plus(s(x), s(y))))) ==> ((plus(s(x), zero) === plus(x, s(zero))) /\
forall(y, plus(x, s(y)) === plus(s(x), y)), forall(y, (plus(x, s(y)) === plus(s(x), y)) ==> (plus(x, s(s(y))) === plus(s(x), s(y))))) ==>
2, forall(y, plus(x, s(y)) === plus(s(x), y)),
Map(sPhi -> LambdaTermFormula(Seq(y), plus(x, s(y)) === plus(s(x), y))) 0,
) Map(sPhi -> LambdaTermFormula(Seq(y), plus(x, s(y)) === plus(s(x), y)))
val inductionPremise4 = RightAnd( )
() |- (plus(x, s(zero)) === plus(s(x), zero)) /\ SCSubproof(SCProof(IndexedSeq(inductionOnY0, inductionInstance1), IndexedSeq(ax"ax7induction")), Seq(-3))
forall(y, (plus(x, s(y)) === plus(s(x), y)) ==> (plus(x, s(s(y))) === plus(s(x), s(y)))), }
Seq(0, 1), val inductionApplication = applyInduction(base0, inductionStep1, inductionInstance)
Seq(plus(x, s(zero)) === plus(s(x), zero), forall(y, (plus(x, s(y)) === plus(s(x), y)) ==> (plus(x, s(s(y))) === plus(s(x), s(y))))) val addForall = RightForall(() |- forall(x, forall(y, plus(x, s(y)) === plus(s(x), y))), inductionApplication.size - 1, forall(y, plus(x, s(y)) === plus(s(x), y)), x)
)
val conclusion5 = hypothesis(forall(y, plus(x, s(y)) === plus(s(x), y)))
val inductionInstanceOnTheLeft6 = LeftImplies(
((plus(s(x), zero) === plus(x, s(zero))) /\
forall(y, (plus(x, s(y)) === plus(s(x), y)) ==> (plus(x, s(s(y))) === plus(s(x), s(y))))) ==>
forall(y, plus(x, s(y)) === plus(s(x), y))
|- forall(y, plus(x, s(y)) === plus(s(x), y)),
4,
5,
(plus(x, s(zero)) === plus(s(x), zero)) /\ forall(y, (plus(x, s(y)) === plus(s(x), y)) ==> (plus(x, s(s(y))) === plus(s(x), s(y)))),
forall(y, plus(x, s(y)) === plus(s(x), y))
)
val inductionResult7 = Cut(
() |- forall(y, plus(x, s(y)) === plus(s(x), y)),
3,
6,
((plus(x, s(zero)) === plus(s(x), zero)) /\ forall(y, (plus(x, s(y)) === plus(s(x), y)) ==> (plus(x, s(s(y))) === plus(s(x), s(y))))) ==> forall(y, plus(x, s(y)) === plus(s(x), y))
)
val addForall8 = RightForall(() |- forall(x, forall(y, plus(x, s(y)) === plus(s(x), y))), 7, forall(y, plus(x, s(y)) === plus(s(x), y)), x)
val proof: SCProof = Proof( val proof: SCProof = Proof(
IndexedSeq(base0, inductionStep1, inductionOnY2, inductionInstance3, inductionPremise4, conclusion5, inductionInstanceOnTheLeft6, inductionResult7, addForall8), inductionApplication :+ addForall,
IndexedSeq(ax"ax3neutral", ax"ax4plusSuccessor", ax"ax7induction") IndexedSeq(ax"ax3neutral", ax"ax4plusSuccessor", ax"ax7induction")
) )
proof proof
......
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