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OneStrictMonotoneSequenceSchema.scala
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OneStrictMonotoneSequenceSchema.scala
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package gapt.examples
import gapt.expr._
import gapt.proofs.context.Context._
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._
object OneStrictMonotoneSequenceSchema extends TacticsProof {
ctx += InductiveType( "nat", hoc"0 : nat", hoc"s : nat>nat" )
ctx += Sort( "i" )
ctx += hoc"f:i>nat"
ctx += hoc"suc:i>i"
ctx += hoc"z:i"
ctx += hoc"E: nat>nat>o"
ctx += hoc"LE: nat>nat>o"
ctx += hoc"Top: nat>nat"
ctx += hoc"omega: nat>nat>nat"
ctx += hoc"nu: nat>nat>i>nat"
ctx += hoc"mu: nat>nat>i>nat"
ctx += hoc"theta: nat>nat>i>nat"
ctx += hoc"chi: nat>i>nat"
ctx += hoc"epsilon: nat>nat>nat>i>nat"
ctx += hoc"delta: nat>nat>i>nat"
ctx += hoc"phi: nat>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 += PrimRecFun( hoc"iNum:nat>i>i", "iNum 0 x = x ", "iNum (s y) x = (suc (iNum y x))" )
ctx += PrimRecFun( hoc"CSeq:nat>nat>i>o", "CSeq 0 n x = (E n (f (iNum 0 x)))", "CSeq (s y) n x = ((E n (f (iNum (s y) x) ) ) ∧ (CSeq y n x))" )
ctx += PrimRecFun( hoc"EndSeq:nat>i>o", "EndSeq 0 x = (E (f x) (f x))", "EndSeq (s y) x = ((E (f x ) (f (iNum (s y) x) )) ∧ (EndSeq y x))" )
// Correct axiom
// ctx += "LEDefinition" -> hos"POR(n,iNum(m,a)) :- LE(f(a), s(n))"
//Incorrect axiom which is inconsistent outside this proof
ctx += "LEDefinitionSingle" -> hos" E(n,f(iNum(m,a))) :- LE(f(a), k)"
ctx += "NumericTransitivity" -> hos"E(n,f(a)),E(n,f(suc(a))) :- E(f(a), f(suc(a)))"
ctx += "NumericTransitivityBase" -> hos"E(n,f(a)) :- E(f(a), f(a))"
ctx += "NumericTransitivityStep" -> hos"E(n,f(iNum(s(k),a))), E(n,f(iNum(k,a))), E(f(a), f(iNum(k,a))) :- E(f(a), f(iNum(s(k),a)))"
ctx += "minimalElement" -> hos"LE(f(z),0) :- "
ctx += "ordcon" -> hos"LE(f(iNum(m,a)),s(n)) :- E(n,f(iNum(m,a))), LE(f(a),n)"
val repeating = le"(s (s 0))"
val esTop = Sequent(
Seq( hof"!x POR(n,x)" ),
Seq( hof"?x ( EndSeq($repeating,x) )" ) )
ctx += ProofNameDeclaration( le"Top n", esTop )
val esOmega = Sequent(
Seq( hof"!x POR(n,x)" ),
Seq( hof"?x ( EndSeq(k,x) )" ) )
ctx += ProofNameDeclaration( le"omega n k", esOmega )
val esPhi = Sequent(
Seq( hof"?x(CSeq(k,n,x)) | !y (LE(f(y),n))" ),
Seq( hof"?x (EndSeq(k,x) )" ) )
ctx += ProofNameDeclaration( le"phi n k", esPhi )
val esMu = Sequent(
Seq( hof"CSeq(k,n,a) " ),
Seq( hof"EndSeq(k,a) " ) )
ctx += ProofNameDeclaration( le"mu k n a", esMu )
val esNu = Sequent(
Seq( hof"!x POR(n,x)" ),
Seq( hof"CSeq(k,n,a)", hof"LE(f(a),n)" ) )
ctx += ProofNameDeclaration( le"nu k n a", esNu )
val esTheta = Sequent(
Seq( hof"!x LE(f(x),s(n))" ),
Seq( hof"LE(f(a),n)", hof"CSeq(k,n,a)" ) )
ctx += ProofNameDeclaration( le"theta k n a", esTheta )
val eschi = Sequent(
Seq( hof" POR(n,a) " ),
Seq( hof"POR(n,a)" ) )
ctx += ProofNameDeclaration( le"chi n a", eschi )
val esEpsilon = Sequent(
Seq( hof" POR(n,iNum(m,a)) " ),
Seq( hof" LE(f(a), k)" ) )
ctx += ProofNameDeclaration( le"epsilon n m k a", esEpsilon )
val esDelta = Sequent(
Seq(
hof" E(n, f(iNum(s(k), a)))",
hof"CSeq(k, n, a)" ),
Seq( hof"E(f(a), f(iNum(s(k), a)))" ) )
ctx += ProofNameDeclaration( le"delta k n a", esDelta )
//The base case of Top
val esTopBc =
Sequent(
Seq( "Ant_0" -> hof"!x POR(0,x)" ),
Seq( "Suc_0" -> hof"?x (EndSeq($repeating ,x))" ) )
val TopBc = Lemma( esTopBc ) {
ref( "omega" )
}
ctx += ProofDefinitionDeclaration( le"Top 0", TopBc )
val esTopSc =
Sequent(
Seq( "Ant_0" -> hof"!x POR(s(n),x)" ),
Seq( "Suc_0" -> hof"?x (EndSeq($repeating ,x))" ) )
val TopSc = Lemma( esTopSc ) {
ref( "omega" )
}
ctx += ProofDefinitionDeclaration( le"Top (s n)", TopSc )
//The base case of omega
val esOmegaBc =
Sequent(
Seq( "Ant_0" -> hof"!x POR(0,x)" ),
Seq( "Suc_0" -> hof"?x (EndSeq(k,x))" ) )
val omegaBc = Lemma( esOmegaBc ) {
cut( "cut", hof"?x (CSeq(k,0,x)) | !y (LE(f(y),0))" )
orR
allR( "cut_1", fov"a" )
exR( "cut_0", fov"a" )
ref( "nu" )
ref( "phi" )
}
ctx += ProofDefinitionDeclaration( le"omega 0 k", omegaBc )
val esOmegaSc =
Sequent(
Seq( "Ant_0" -> hof"!x POR(s(n),x)" ),
Seq( "Suc_0" -> hof"?x (EndSeq(k,x))" ) )
val omegaSc = Lemma( esOmegaSc ) {
cut( "cut", hof"?x (CSeq(k,s(n),x)) | !y (LE(f(y),s(n)))" )
orR
allR( "cut_1", fov"a" )
exR( "cut_0", fov"a" )
ref( "nu" )
ref( "phi" )
}
ctx += ProofDefinitionDeclaration( le"omega (s n) k", omegaSc )
val esPhiBc1 =
Sequent(
Seq( "Ant_0" -> hof"?x (CSeq(0,0,x)) | !y (LE(f(y),0))" ),
Seq( "Suc_0" -> hof"?x (EndSeq(0,x) )" ) )
val phiBc1 = Lemma( esPhiBc1 ) {
orL
exL( fov"a" )
exR( fov"a" )
unfold( "CSeq" ) atMost 1 in "Ant_0"
unfold( "iNum" ) atMost 1 in "Ant_0"
unfold( "EndSeq" ) atMost 1 in "Suc_0_0"
ref( "NumericTransitivityBase" )
allL( foc"z" )
ref( "minimalElement" )
}
ctx += ProofDefinitionDeclaration( le"phi 0 0", phiBc1 )
val esPhiBc2 =
Sequent(
Seq( "Ant_0" -> hof"?x (CSeq(s(k),0,x)) | !y (LE(f(y),0))" ),
Seq( "Suc_0" -> hof"?x (EndSeq(s(k),x) )" ) )
val phiBc2 = Lemma( esPhiBc2 ) {
orL
exL( fov"a" )
exR( fov"a" )
unfold( "CSeq" ) atMost 1 in "Ant_0"
unfold( "EndSeq" ) atMost 1 in "Suc_0_0"
andL
andR
ref( "delta" )
ref( "mu" )
allL( foc"z" )
ref( "minimalElement" )
}
ctx += ProofDefinitionDeclaration( le"phi 0 (s k)", phiBc2 )
val esPhiSc1 =
Sequent(
Seq( "Ant_0" -> hof"?x (CSeq(0,s(n),x)) | !y (LE(f(y),s(n)))" ),
Seq( "Suc_0" -> hof"?x (EndSeq(0,x) )" ) )
val phiSc1 = Lemma( esPhiSc1 ) {
cut( "cut", hof"?x(CSeq(0,n,x)) | !y (LE(f(y),n))" )
orR
orL
exL( "Ant_0", fov"a" )
exR( "Suc_0", fov"a" )
unfold( "CSeq" ) atMost 1 in "Ant_0"
unfold( "EndSeq" ) atMost 1 in "Suc_0_0"
unfold( "iNum" ) atMost 1 in "Ant_0"
ref( "NumericTransitivityBase" )
allR( fov"b" )
exR( "cut_0", fov"b" )
allL( le"(iNum 0 b)" )
unfold( "CSeq" ) atMost 1 in "cut_0_0"
ref( "ordcon" )
ref( "phi" )
}
ctx += ProofDefinitionDeclaration( le"phi (s n) 0", phiSc1 )
val esPhiSc2 =
Sequent(
Seq( "Ant_0" -> hof"?x (CSeq(s(k),s(n),x)) | !y (LE(f(y),s(n)))" ),
Seq( "Suc_0" -> hof"?x (EndSeq(s(k),x) )" ) )
val phiSc2 = Lemma( esPhiSc2 ) {
cut( "cut", hof"?x(CSeq(s(k),n,x)) | !y (LE(f(y),n))" )
orR
orL
exL( "Ant_0", fov"a" )
exR( "Suc_0", fov"a" )
unfold( "CSeq" ) atMost 1 in "Ant_0"
unfold( "EndSeq" ) atMost 1 in "Suc_0_0"
andL
andR
ref( "delta" )
ref( "mu" )
allR( fov"b" )
exR( "cut_0", fov"b" )
allL( le"(iNum (s k) b)" )
unfold( "CSeq" ) atMost 1 in "cut_0_0"
andR
ref( "ordcon" )
ref( "theta" )
ref( "phi" )
}
ctx += ProofDefinitionDeclaration( le"phi (s n) (s k)", phiSc2 )
val esmuBc =
Sequent(
Seq( "Ant_0" -> hof"CSeq(0,n,a)" ),
Seq( "Suc_0" -> hof"EndSeq(0,a)" ) )
val muBc = Lemma( esmuBc ) {
unfold( "CSeq" ) atMost 1 in "Ant_0"
unfold( "EndSeq" ) atMost 1 in "Suc_0"
unfold( "iNum" ) atMost 1 in "Suc_0"
unfold( "iNum" ) atMost 1 in "Ant_0"
ref( "NumericTransitivityBase" )
}
ctx += ProofDefinitionDeclaration( le"mu 0 n a", muBc )
val esmuSc =
Sequent(
Seq( "Ant_0" -> hof"CSeq(s(k),n,a)" ),
Seq( "Suc_0" -> hof"EndSeq(s(k),a)" ) )
val muSc = Lemma( esmuSc ) {
unfold( "CSeq" ) atMost 1 in "Ant_0"
unfold( "EndSeq" ) atMost 1 in "Suc_0"
andL
andR
ref( "delta" )
ref( "mu" )
}
ctx += ProofDefinitionDeclaration( le"mu (s k) n a", muSc )
val esThetaBc =
Sequent(
Seq( "Ant_0" -> hof"!x LE(f(x),s(n))" ),
Seq(
"Suc_0" -> hof"CSeq(0,n,a)",
"Suc_1" -> hof"LE(f(a),n)" ) )
val thetaBc = Lemma( esThetaBc ) {
unfold( "CSeq" ) atMost 1 in "Suc_0"
allL( le"(iNum 0 a)" )
ref( "ordcon" )
}
ctx += ProofDefinitionDeclaration( le"theta 0 n a", thetaBc )
val esThetaSc =
Sequent(
Seq( "Ant_0" -> hof"!x LE(f(x),s(n))" ),
Seq(
"Suc_0" -> hof"CSeq(s(k),n,a)",
"Suc_1" -> hof"LE(f(a),n)" ) )
val thetaSc = Lemma( esThetaSc ) {
unfold( "CSeq" ) atMost 1 in "Suc_0"
andR
allL( le"(iNum (s k) a)" )
ref( "ordcon" )
ref( "theta" )
}
ctx += ProofDefinitionDeclaration( le"theta (s k) n a", thetaSc )
// The Basecase of chi
val esChiBc = Sequent(
Seq(
"Ant_2" -> hof" POR(0,a)" ),
Seq(
"Suc_0" -> hof"POR(0,a)" ) )
val chiBc = Lemma( esChiBc ) {
unfold( "POR" ) atMost 1 in "Suc_0"
unfold( "POR" ) atMost 1 in "Ant_2"
trivial
}
ctx += ProofDefinitionDeclaration( le"chi 0 a", chiBc )
//The step case of chi
val esChiSc = Sequent(
Seq(
"Ant_2" -> hof" POR(s(n),a)" ),
Seq(
"Suc_0" -> hof"POR(s(n),a)" ) )
val chiSc = Lemma( esChiSc ) {
unfold( "POR" ) atMost 1 in "Suc_0"
unfold( "POR" ) atMost 1 in "Ant_2"
orR
orL
trivial
ref( "chi" )
}
ctx += ProofDefinitionDeclaration( le"chi (s n) a", chiSc )
val esEpsilonBc = Sequent(
Seq(
"Ant_2" -> hof" POR(0,iNum(m,a))" ),
Seq(
"Suc_0" -> hof"LE(f(a), k)" ) )
val epsilonBc = Lemma( esEpsilonBc ) {
unfold( "POR" ) atMost 1 in "Ant_2"
unfold( "POR" ) atMost 1 in "Ant_2"
ref( "LEDefinitionSingle" )
}
ctx += ProofDefinitionDeclaration( le"epsilon 0 m k a", epsilonBc )
val esEpsilonSc = Sequent(
Seq(
"Ant_2" -> hof" POR(s(n),iNum(m,a))" ),
Seq(
"Suc_0" -> hof"LE(f(a), k)" ) )
val epsilonSc = Lemma( esEpsilonSc ) {
unfold( "POR" ) atMost 1 in "Ant_2"
orL
ref( "LEDefinitionSingle" )
ref( "epsilon" )
}
ctx += ProofDefinitionDeclaration( le"epsilon (s n) m k a", epsilonSc )
val esDeltaBc = Sequent(
Seq(
"Ant_0" -> hof" E(n, f(iNum(s(0), a)))",
"Ant_1" -> hof"CSeq(0, n, a)" ),
Seq(
"Suc_0" -> hof"E(f(a), f(iNum(s(0), a)))" ) )
val deltaBc = Lemma( esDeltaBc ) {
unfold( "CSeq" ) atMost 1 in "Ant_1"
unfold( "iNum" ) atMost 1 in "Ant_1"
unfold( "iNum" ) in "Suc_0"
unfold( "iNum" ) in "Ant_0"
ref( "NumericTransitivity" )
}
ctx += ProofDefinitionDeclaration( le"delta 0 n a", deltaBc )
val esDeltaSc = Sequent(
Seq(
"Ant_0" -> hof" E(n, f(iNum(s(s(k)), a)))",
"Ant_1" -> hof"CSeq(s(k), n, a)" ),
Seq(
"Suc_0" -> hof"E(f(a), f(iNum(s(s(k)), a)))" ) )
val deltaSc = Lemma( esDeltaSc ) {
unfold( "CSeq" ) atMost 1 in "Ant_1"
andL
cut( "cut", hof"E(f(a), f(iNum(s(k), a)))" )
ref( "delta" )
ref( "NumericTransitivityStep" )
}
ctx += ProofDefinitionDeclaration( le"delta (s k) n a", deltaSc )
}