Assignment 11 (1)
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U N I V E R S I T A S S A R A V I E N S I S Assignment 11 Introduction to Computational Logic, SS 2015 Prof. Dr. Gert Smolka, Tobias Tebbi https://courses.ps.uni-saarland.de/icl_15/ Read Chapter 10 of the lecture notes. Do all the proofs on this sheet on paper as well as in Coq. Exercise 11.1 Prove the following with diagrams on paper and with Coq. You may use the following admissible rules. Weak A ‘ s B ‘ s A ⊆ B G A ‘ C s A ‘ ¬¬s IC A ‘ s A ‘ C s a) s ‘ t → A ‘ s → A ‘ t b) s,t ‘ u → A ‘ s → A ‘ t → A ‘ u c) A ‘ ¬¬s → A ‘ C s d) s ‘ ¬¬s e) ¬¬s ‘ C s f) A ‘ C s ↔ A, ¬s ‘ C ⊥ g) A ‘ s → t ↔ A, s ‘ t h) A ‘ s → t ↔ (∀A 0 . A ⊆ A 0 → A 0 ‘ s → A 0 ‘ t) i) ¬¬⊥ ‘ ⊥ Exercise 11.2 Prove that A ‘ s → A ‘ C s . Exercise 11.3 Prove weakening for classical ND. A ‘ C s B ‘ C s A ⊆ B Exercise 11.4 Prove that the refutation lemma (A ‘⊥↔ A ‘ C ⊥) is equivalent to Glivenko’s theorem. Exercise 11.5 Prove the following variant of Glivenko’s theorem by induction. A ‘ C s → A, ¬s ‘⊥ Exercise 11.6 Prove that the following rules are admissible using a diagram and in Coq. A ‘ s s ‘ t A ‘ t A ‘ C s A ‘ C u s → u ∈ A Exercise 11.7 Let A be a context. Use Glivenko’s Theorem to prove there is no formula s such that A ‘ C ¬s and A 6‘¬s . 2015–07–10 00:56
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Transcript of Assignment 11 (1)
UN
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Assignment 11Introduction to Computational Logic, SS 2015
Prof. Dr. Gert Smolka, Tobias Tebbi
https://courses.ps.uni-saarland.de/icl_15/
Read Chapter 10 of the lecture notes.
Do all the proofs on this sheet on paper as well as in Coq.
Exercise 11.1 Prove the following with diagrams on paper and with Coq. You
may use the following admissible rules.
WeakA ` sB ` s
A ⊆ B GA `CsA ` ¬¬s
ICA ` sA `Cs
a) s ` t → A ` s → A ` tb) s, t ` u → A ` s → A ` t → A ` uc) A ` ¬¬s → A `Csd) s ` ¬¬se) ¬¬s `Csf) A `Cs ↔ A,¬s `C⊥g) A ` s → t ↔ A, s ` th) A ` s → t ↔ (∀A′. A ⊆ A′ → A′ ` s → A′ ` t)i) ¬¬⊥ ` ⊥
Exercise 11.2 Prove that A ` s → A `Cs.
Exercise 11.3 Prove weakening for classical ND.
A `CsB `Cs
A ⊆ B
Exercise 11.4 Prove that the refutation lemma (A ` ⊥ ↔ A `C⊥) is equivalent to
Glivenko’s theorem.
Exercise 11.5 Prove the following variant of Glivenko’s theorem by induction.
A `Cs → A,¬s ` ⊥
Exercise 11.6 Prove that the following rules are admissible using a diagram and
in Coq.
A ` s s ` tA ` t
A `CsA `Cu
s → u ∈ A
Exercise 11.7 Let A be a context. Use Glivenko’s Theorem to prove there is no
formula s such that A `C¬s and A 6` ¬s.
2015–07–10 00:56
Recall the Hilbert system for intuitionistic propositional logic
KH s → t → s
SH (s → t → u)→ (s → t)→ s → u
EH⊥→ u
MPH s → t H s
H t
and its variant with assumptions
CA `Hs
s ∈ A KA `Hs → t → s
SA `H(s → t → u)→ (s → t)→ s → u
EA `H⊥ → u
MPA `Hs → t A `Hs
A `Ht
Exercise 11.8 ProveH (s → s) as a diagram on paper and in Coq. Hint: “SKK”.
Exercise 11.9 Prove
A ` s → A `Hs
assuming
D : ∀A, s, t. A, s `Ht → A `Hs → t
Exercise 11.10 Prove the following form of weakening for the Hilbert calculus.
A `Ht → A, s `Ht
Exercise 11.11 (Substitutivity)
Define a substitution function subst : (var→ form)→ form→ form such that
substσ s is s where every variable x is replaced by σx. We write σ s for substσ s.Prove the following.
SubstitutivityA ` s
σ A ` σ s
Exercise 11.12 Consider formulas s, t ::= x | s ∨ t and define A `Ds such that
A `Ds ∨ t ↔ ∀u. A, s `Du → A, t `Du → A `Du
a) Define `D with the context rule, two introduction rules for ∨, and one elimi-
nation rule for ∨.
b) Prove the characteristic equivalence for ∨ given above.
c) Profe that `D satisfies weakening.
Exercise 11.13 (Challenge) In this exercise, we show that the following attempt
to formulate a characteristic property for implication is wrong.
A ` s → t ↔ (A ` s → A ` t)
That is, prove that
¬∀As t. A ` s → t ↔ (A ` s → A ` t)
You may assume that for every variable x, we have 6` x and 6` ¬x, that is, neither
x nor its negation are provable.
2015–07–10 00:56
