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EventuallyConstantSchema.scala
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EventuallyConstantSchema.scala
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package gapt.examples
import gapt.expr._
import gapt.proofs.Sequent
import gapt.proofs.context.Context
import gapt.proofs.context.update.InductiveType
import gapt.proofs.context.update.{ PrimitiveRecursiveFunction => PrimRecFun }
import gapt.proofs.context.update.ProofDefinitionDeclaration
import gapt.proofs.context.update.ProofNameDeclaration
import gapt.proofs.context.update.Sort
import gapt.proofs.gaptic.TacticsProof
import gapt.proofs.gaptic._
object EventuallyConstantSchema extends TacticsProof {
ctx += InductiveType( "nat", hoc"0 : nat", hoc"s : nat>nat" )
ctx += Sort( "i" )
ctx += hoc"f:i>nat"
ctx += hoc"g:i>i"
ctx += hoc"z:i"
ctx += hoc"E: nat>nat>o"
ctx += hoc"LE: nat>nat>o"
ctx += hoc"LEQ: nat>nat>o"
ctx += hoc"iLEQ: i>i>o"
ctx += hoc"omega: nat>nat"
ctx += hoc"phi: nat>nat"
ctx += PrimRecFun( hoc"POR:nat>i>o", "POR 0 x = E 0 (f x) ", "POR (s y) x = (E (s y) (f x) ∨ POR y x)" )
ctx += "UpBound" -> hos"POR(n,a) :- LE(f(a), s(n))"
ctx += "gEq" -> hos"E(n,f(a)),E(n,f(g(a))) :- E(f(a), f(g(a)))"
ctx += "smallest" -> hos"LE(n,0) :- "
ctx += "reflex" -> hos" :- iLEQ(a,a)"
ctx += "ordcon" -> hos"LE(f(a),s(n)),iLEQ(a,b) :- E(n,f(b)), LE(f(b),n)"
val esOmega = Sequent(
Seq( hof"!x POR(n,x)" ),
Seq( hof"?x (iLEQ(x,g(x)) -> E(f(x), f(g(x))) )" ) )
ctx += ProofNameDeclaration( le"omega n", esOmega )
val esPhi = Sequent(
Seq( hof"?x !y ((iLEQ(x,y) -> E(n,f(y))) | LE(f(y),n))" ),
Seq( hof"?x (iLEQ(x,g(x)) -> E(f(x), f(g(x))) )" ) )
ctx += ProofNameDeclaration( le"phi n", esPhi )
//The base case of omega
val esOmegaBc =
Sequent(
Seq( "Ant_0" -> hof"!x POR(0,x)" ),
Seq( "Suc_0" -> hof"?x (iLEQ(x,g(x)) -> E(f(x), f(g(x))))" ) )
val omegaBc = Lemma( esOmegaBc ) {
cut( "cut", hof"?x !y ((iLEQ(x,y) -> E(0,f(y))) | LE(f(y),0))" ) right ref( "phi" )
exR( "cut", hoc"z" )
allR( "cut_0", fov"a" )
orR
impR
allL( "Ant_0", fov"a" )
unfold( "POR" ) atMost 1 in "Ant_0_0"
trivial
}
ctx += ProofDefinitionDeclaration( le"omega 0", omegaBc )
val esOmegaSc =
Sequent(
Seq( "Ant_0" -> hof"!x POR(s(n),x)" ),
Seq( "Suc_0" -> hof"?x (iLEQ(x,g(x)) -> E(f(x), f(g(x))))" ) )
val omegaSc = Lemma( esOmegaSc ) {
cut( "cut", hof"?x !y ((iLEQ(x,y) -> E(s(n),f(y))) | LE(f(y),s(n)))" ) right ref( "phi" )
exR( "cut", hoc"z" )
allR( "cut_0", fov"a" )
orR
impR
allL( "Ant_0", fov"a" )
unfold( "POR" ) atMost 1 in "Ant_0_0"
orL left trivial; foTheory
}
ctx += ProofDefinitionDeclaration( le"omega (s n)", omegaSc )
val esPhiBc =
Sequent(
Seq( "Ant_0" -> hof"?x !y ((iLEQ(x,y) -> E(0,f(y))) | LE(f(y),0))" ),
Seq( "Suc_0" -> hof"?x (iLEQ(x,g(x)) -> E(f(x), f(g(x))) )" ) )
val phiBc = Lemma( esPhiBc ) {
exL( fov"a" )
allL( fov"a" )
allL( le"(g a)" )
exR( fov"a" )
impR
orL( "Ant_0_0" ) right foTheory
orL( "Ant_0_1" ) right foTheory
impL( "Ant_0_0" )
by {
impL( "Ant_0_1" ) left foTheory
foTheory
}
by {
impL( "Ant_0_1" ) left trivial
foTheory
}
}
ctx += ProofDefinitionDeclaration( le"phi 0", phiBc )
val esPhiSc =
Sequent(
Seq( "Ant_0" -> hof"?x !y ((iLEQ(x,y) -> E(s(n),f(y))) | LE(f(y),s(n)))" ),
Seq( "Suc_0" -> hof"?x (iLEQ(x,g(x)) -> E(f(x), f(g(x))) )" ) )
val phiSc = Lemma( esPhiSc ) {
cut( "cut", hof"?x !y ((iLEQ(x,y) -> E(n,f(y))) | LE(f(y),n))" ) right ref( "phi" )
cut( "cut1", hof"?x !y ( iLEQ(x,y) -> E(s(n),f(y)) )" ) left by {
cut( "cut2", hof"?x ( LE(f(x),s(n)) )" ) left by {
forget( "cut" )
exL( fov"a" )
exR( "cut1", fov"a" )
allR( "cut1_0", fov"b" )
allL( fov"b" )
exR( "cut2", fov"b" )
prop
}
by {
exL( "cut2", fov"a" )
exR( "cut", fov"a" )
allR( fov"b" )
orR
impR
foTheory
}
}
exL( "cut1", fov"a" )
allL( fov"a" )
allL( le"(g a)" )
exR( "Suc_0", fov"a" )
impL( "cut1_1" ) left {
impL onAll by { impR; trivial }
}
impL( "cut1_0" ) left foTheory
impR
foTheory
}
ctx += ProofDefinitionDeclaration( le"phi (s n)", phiSc )
}