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examples.scala
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examples.scala
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package gapt.examples
import gapt.expr._
import gapt.proofs.Sequent
import gapt.proofs.gaptic._
object gapticExamples {
val decomposeProof = Proof(
( "label1" -> hof"!x (p x & q x)" ) +:
Sequent()
:+ ( "label2" -> hof"!y (p y -> q y | r y)" ) ) {
decompose
allL( fov"y" )
decompose
trivial
}
val destructProof = Proof(
( "label1" -> hof"a & (b & c)" ) +:
Sequent()
:+ ( "label2" -> hof"a | (b | c)" ) ) {
destruct( "label1" )
destruct( "label2" )
trivial
}
val destructProof2 = Proof(
( "noise1" -> hof"a" ) +:
Sequent()
:+ ( "noise2" -> hof"P(y)" )
:+ ( "label1" -> hof"a | (b | c)" )
:+ ( "noise3" -> hof"P(z)" )
:+ ( "label2" -> hof"a & (b & c)" ) ) {
destruct( "label1" )
destruct( "label2" )
trivial
trivial
}
val chainProof = Proof(
( "a" -> hof"q(f(c))" ) +:
( "hyp" -> hof"!x (q x -> p (f x))" ) +:
Sequent()
:+ ( "target" -> hof"p(f(f(c)))" ) ) {
chain( "hyp" ).at( "target" )
trivial
}
val chainProof2 = Proof(
( "hyp" -> hof"!x p (f x)" ) +:
Sequent()
:+ ( "target" -> hof"p(f(f(c)))" ) ) {
chain( "hyp" )
}
val chainProof3 = Proof(
( "a" -> hof"r(f(c))" ) +:
( "b" -> hof"q(f(c))" ) +:
( "hyp" -> hof"!x (r x -> q x -> p (f x))" ) +:
Sequent()
:+ ( "target" -> hof"p(f(f(c)))" ) ) {
chain( "hyp" )
trivial
trivial
}
val chainProof4 = Proof(
( "a" -> hof"r(f(c))" ) +:
( "b" -> hof"q(f(c))" ) +:
( "c" -> hof"w(f(c))" ) +:
( "hyp" -> hof"!x (r x & q x & w x -> p (f x))" ) +:
Sequent()
:+ ( "target" -> hof"p(f(f(c)))" ) ) {
chain( "hyp" )
trivial
trivial
trivial
}
val eqProof = Proof(
( "c" -> hof"P(y) & Q(y)" ) +:
( "eq1" -> hof"u = v" ) +:
( "eq2" -> hof"y = x" ) +:
( "a" -> hof"P(u) -> Q(u)" ) +:
Sequent()
:+ ( "b" -> hof"P(x) & Q(x)" ) ) {
eql( "eq1", "a" ) yielding hof"P(v) -> Q(v)"
eql( "eq1", "a" ) yielding hof"P(v) -> Q(u)"
eql( "eq2", "b" ).fromRightToLeft
trivial
}
val lemma = Proof(
( "A" -> hof"A -> B" ) +:
Sequent()
:+ ( "S" -> hof"A & B | -A" ) ) {
orR
negR
andR
repeat( trivial )
impL
repeat( trivial )
}
val lemma2 = Proof(
( "A" -> hof"A -> B" ) +:
Sequent()
:+ ( "S" -> hof"A & B | -A" ) ) {
repeat( orR orElse negR orElse andR orElse impL orElse trivial )
}
val drinker3 = Proof( Sequent()
:+ ( "E" -> hof"B" )
:+ ( "E" -> hof"A" )
:+ ( "D" -> hof"?x (P x -> !y P y)" ) ) {
exR( le"c" )
impR
allR
exR( le"y" )
impR
allR
trivial
}
val lemma3 = Proof(
( "F" -> hof"A -> B" ) +:
Sequent()
:+ ( "E" -> hof"B" )
:+ ( "D" -> hof"?y (P y -> !z P z)" ) ) {
impL
insert( drinker3 )
trivial
}
val lemma_ = Proof(
( "initAnt" -> hof"A -> B" ) +:
Sequent()
:+ ( "initSuc" -> hof"A & B | -A" ) ) {
orR( "initSuc" )
negR( "initSuc_1" )
andR( "initSuc_0" )
axiomLog
impL( "initAnt" )
axiomLog
axiomLog
}
val lemma2_ = Proof(
( "initAnt" -> hof"A -> B" ) +:
Sequent()
:+ ( "initSuc" -> hof"A & B | -A" ) ) {
orR( "initSuc" )
negR( "initSuc_1" )
andR( "initSuc_0" )
trivial
impL
trivial
trivial
}
val direct = Proof(
( "A" -> hof"A" ) +: ( "B" -> hof"B" ) +: Sequent() :+ ( "B_" -> hof"B" ) ) {
trivial
}
val lemmaProp = Proof(
( "a" -> hof"A -> B" ) +: Sequent() :+ ( "s" -> hof"A&B | -A" ) ) {
impL
prop
prop
}
}