Proving Decidability of Intuitionistic Propositional Calculus on Coq

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直観主義命題論理の決定性をCoqで証明した話 @ Logic Zoo Workshop 2013 http://logiczoo13.pira.jp/

直観主義命題論理の決定性をCoqで証明した話 @ Logic Zoo Workshop 2013 http://logiczoo13.pira.jp/

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  • 1. Proving decidabilityof Intuitionistic Propositional Calculus on Coq Masaki Hara (qnighy) University of Tokyo, first grade Logic Zoo 2013 にて
  • 2. 1. Task & Known results2. Brief methodology of the proof 1. Cut elimination 2. Contraction elimination 3. → 𝐿 elimination 4. Proof of strictly-decreasingness3. Implementation detail4. Further implementation plan
  • 3. Task• Proposition: 𝐴𝑡𝑜𝑚 𝑛 , ∧, ∨, →, ⊥• Task: Is given propositional formula P provable in LJ? – It’s known to be decidable. [Dyckhoff]• This talk: how to prove this decidability on Coq
  • 4. Known results• Decision problem on IPC is PSPACE complete [Statman] – Especially, O(N log N) space decision procedure is known [Hudelmaier]• These approaches are backtracking on LJ syntax.
  • 5. Known results• cf. classical counterpart of this problem is co-NP complete. – Proof: find counterexample in boolean-valued semantics (SAT).
  • 6. methodology• To prove decidability, all rules should be strictly decreasing on some measuring. 𝑆1 ,𝑆2 ,…,𝑆 𝑁• More formally, for all rules 𝑟𝑢𝑙𝑒 𝑆0 and all number 𝑖 (1 ≤ 𝑖 ≤ 𝑁), 𝑆 𝑖 < 𝑆0 on certain well-founded relation <.
  • 7. methodology1. Eliminate cut rule of LJ2. Eliminate contraction rule3. Split → 𝑳 rule into 4 pieces4. Prove that every rule is strictly decreasing
  • 8. Sequent Calculus LJ Γ⊢𝐺 𝐴,𝐴,Γ⊢𝐺 Γ⊢𝐴 𝐴,Δ⊢𝐺• 𝑤𝑒𝑎𝑘 𝑐𝑜𝑛𝑡𝑟 (𝑐𝑢𝑡) 𝐴,Γ⊢𝐺 𝐴,Γ⊢𝐺 Γ,Δ⊢𝐺• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴⊢𝐴 ⊥⊢𝐺 Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵•
  • 9. Sequent Calculus LJ Γ⊢𝐺 𝐴,𝐴,Γ⊢𝐺 Γ⊢𝐴 𝐴,Δ⊢𝐺• 𝑤𝑒𝑎𝑘 𝑐𝑜𝑛𝑡𝑟 (𝑐𝑢𝑡) 𝐴,Γ⊢𝐺 𝐴,Γ⊢𝐺 Γ,Δ⊢𝐺• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴⊢𝐴 ⊥⊢𝐺 Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵• We eliminate cut rule first.
  • 10. Cut elimination• 1. Prove these rule by induction on proof structure. Γ⊢𝐺 Δ,Δ,Γ⊢𝐺• 𝑤𝑒𝑎𝑘𝐺 𝑐𝑜𝑛𝑡𝑟𝐺 Δ,Γ⊢𝐺 Δ,Γ⊢𝐺 Γ⊢⊥• ⊥ 𝑅𝐸 Γ⊢𝐺 Γ⊢𝐴∧𝐵 Γ⊢𝐴∧𝐵• ∧ 𝑅𝐸1 ∧ 𝑅𝐸2 Γ⊢𝐴 Γ⊢𝐵 Γ⊢𝐴→𝐵• → 𝑅𝐸 𝐴,Γ⊢𝐵 Γ1 ⊢𝐴 𝐴,Δ1 ⊢𝐺1 Γ2 ⊢𝐵 𝐵,Δ2 ⊢𝐺2• If (𝑐𝑢𝑡 𝐴 ) and (𝑐𝑢𝑡 𝐵 ) for all Γ1 ,Δ1 ⊢𝐺1 Γ2 ,Δ2 ⊢𝐺2 Γ⊢𝐴∨𝐵 A,Δ⊢𝐺 𝐵,Δ⊢𝐺 Γ1 , Γ2 , Δ1 , Δ2 , 𝐺1 , 𝐺2 , then (∨ 𝑅𝐸 ) Γ,Δ⊢𝐺
  • 11. Cut elimination• 2. Prove the general cut rule Γ ⊢ 𝐴 𝐴 𝑛 , Δ ⊢ 𝐺 𝑐𝑢𝑡𝐺 Γ, Δ ⊢ 𝐺 by induction on the size of 𝐴 and proof structure of the right hand.• 3. specialize 𝑐𝑢𝑡𝐺 (n = 1) ■
  • 12. Cut-free LJ Γ⊢𝐺 𝐴,𝐴,Γ⊢𝐺• 𝑤𝑒𝑎𝑘 𝑐𝑜𝑛𝑡𝑟 𝐴,Γ⊢𝐺 𝐴,Γ⊢𝐺• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴⊢𝐴 ⊥⊢𝐺 Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵•
  • 13. Cut-free LJ Γ⊢𝐺 𝐴,𝐴,Γ⊢𝐺• 𝑤𝑒𝑎𝑘 𝑐𝑜𝑛𝑡𝑟 𝐴,Γ⊢𝐺 𝐴,Γ⊢𝐺• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴⊢𝐴 ⊥⊢𝐺 Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵• Contraction rule is not strictly decreasing
  • 14. Contraction-free LJ• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺 𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵
  • 15. Contraction-free LJ• Implicit weak – 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺• Implicit contraction 𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺 – →𝐿 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵 – (∧ 𝑅 ) Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 – ∨𝐿 𝐴∨𝐵,Γ⊢𝐺
  • 16. Contraction-free LJ• Implicit weak – 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺• Implicit contraction 𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺 – →𝐿 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵 – (∧ 𝑅 ) Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 – ∨𝐿 𝐴∨𝐵,Γ⊢𝐺
  • 17. Proof of weak rule• Easily done by induction ■
  • 18. Proof of contr rule• 1. prove these rules by induction on proof structure. 𝐴∧𝐵,Γ⊢𝐺 𝐴∨𝐵,Γ⊢𝐺 𝐴∨𝐵,Γ⊢𝐺 – ∧ 𝐿𝐸 ∨ 𝐿𝐸1 (∨ 𝐿𝐸2 ) 𝐴,𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 𝐴→𝐵,Γ⊢𝐺 – (→ 𝑤𝑒𝑎𝑘 ) 𝐵,Γ⊢𝐺• 2. prove contr rule by induction on proof structure.■
  • 19. Contraction-free LJ• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺 𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵
  • 20. Contraction-free LJ• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺 𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵• This time, → 𝐿 rule is not decreasing
  • 21. Terminating LJ 𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺• Split →𝐿 into 4 pieces 𝐴→𝐵,Γ⊢𝐺 𝐶,𝐴𝑡𝑜𝑚 𝑛 ,Γ⊢𝐺 1. → 𝐿1 𝐴𝑡𝑜𝑚 𝑛 →𝐶,𝐴𝑡𝑜𝑚 𝑛 ,Γ⊢𝐺 𝐵→𝐶,Γ⊢𝐴→𝐵 C,Γ⊢𝐺 2. (→ 𝐿2 ) 𝐴→𝐵 →𝐶,Γ⊢𝐺 𝐴→ 𝐵→𝐶 ,Γ⊢𝐺 3. (→ 𝐿3 ) 𝐴∧𝐵 →𝐶,Γ⊢𝐺 𝐴→𝐶,𝐵→𝐶,Γ⊢𝐺 4. (→ 𝐿4 ) 𝐴∨𝐵 →𝐶,Γ⊢𝐺
  • 22. Correctness of Terminating LJ• 1. If Γ ⊢ 𝐺 is provable in Contraction-free LJ, At least one of these is true: – Γ includes ⊥, 𝐴 ∧ 𝐵, or 𝐴 ∨ 𝐵 – Γ includes both 𝐴𝑡𝑜𝑚(𝑛) and 𝐴𝑡𝑜𝑚 𝑛 → 𝐵 – Γ ⊢ 𝐺 has a proof whose bottommost rule is not the form of 𝐴𝑡𝑜𝑚 𝑛 →𝐵,𝐴𝑡𝑜𝑚 𝑛 ,Γ⊢𝐴𝑡𝑜𝑚 𝑛 𝐵,𝐴𝑡𝑜𝑚 𝑛 ,Γ⊢𝐺 (→ 𝐿 ) 𝐴𝑡𝑜𝑚 𝑛 →𝐵,𝐴𝑡𝑜𝑚(𝑛),Γ⊢𝐺• Proof: induction on proof structure
  • 23. Correctness of Terminating LJ• 2. every sequent provable in Contraction-free LJ is also provable in Terminating LJ.• Proof: induction by size of the sequent. – Size: we will introduce later
  • 24. Terminating LJ• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺 𝐶,𝐴𝑡𝑜𝑚 𝑛 ,Γ⊢𝐺 𝐵→𝐶,Γ⊢𝐴→𝐵 C,Γ⊢𝐺• → 𝐿1 → 𝐿2 𝐴𝑡𝑜𝑚 𝑛 →𝐶,𝐴𝑡𝑜𝑚 𝑛 ,Γ⊢𝐺 𝐴→𝐵 →𝐶,Γ⊢𝐺 𝐴→ 𝐵→𝐶 ,Γ⊢𝐺 𝐴→𝐶,𝐵→𝐶,Γ⊢𝐺• → 𝐿3 → 𝐿4 𝐴∧𝐵 →𝐶,Γ⊢𝐺 𝐴∨𝐵 →𝐶,Γ⊢𝐺 𝐴,Γ⊢𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• →𝑅 ∧𝐿 (∧ 𝑅 ) Γ⊢𝐴→𝐵 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵
  • 25. Proof of termination• Weight of Proposition – 𝑤 𝐴𝑡𝑜𝑚 𝑛 = 1 – 𝑤 ⊥ =1 – 𝑤 𝐴 → 𝐵 = 𝑤 𝐴 + 𝑤 𝐵 +1 – 𝑤 𝐴∧ 𝐵 = 𝑤 𝐴 + 𝑤 𝐵 +2 – 𝑤 𝐴∨ 𝐵 = 𝑤 𝐴 + 𝑤 𝐵 +1• 𝐴 < 𝐵 ⇔ 𝑤 𝐴 < 𝑤(𝐵)
  • 26. Proof of termination• ordering of Proposition List – Use Multiset ordering (Dershowitz and Manna ordering)
  • 27. Multiset Ordering• Multiset Ordering: a binary relation between multisets (not necessarily be ordering)• 𝐴> 𝐵⇔ Not empty A B
  • 28. Multiset Ordering• If 𝑅 is a well-founded binary relation, the Multiset Ordering over 𝑅 is also well-founded.• Well-founded: every element is accessible• 𝐴 is accessible : every element 𝐵 such that 𝐵 < 𝐴 is accessible
  • 29. Multiset OrderingProof• 1. induction on list• Nil ⇒ there is no 𝐴 such that 𝐴 < 𝑀 Nil, therefore it’s accessible.• We will prove: 𝐴𝑐𝑐 𝑀 𝐿 ⇒ 𝐴𝑐𝑐 𝑀 (𝑥 ∷ 𝐿)
  • 30. Multiset Ordering• 2. duplicate assumption• Using 𝐴𝑐𝑐(𝑥) and 𝐴𝑐𝑐 𝑀 (𝐿), we will prove 𝐴𝑐𝑐 𝑀 𝐿 ⇒ 𝐴𝑐𝑐 𝑀 (𝑥 ∷ 𝐿)• 3. induction on 𝑥 and 𝐿 – We can use these two inductive hypotheses. 1. ∀𝐾 𝑦, 𝑦 < 𝑥 ⇒ 𝐴𝑐𝑐 𝑀 𝐾 ⇒ 𝐴𝑐𝑐 𝑀 (𝑦 ∷ 𝐾) 2. ∀𝐾, 𝐾 < 𝑀 𝐿 ⇒ 𝐴𝑐𝑐 𝑀 𝐾 ⇒ 𝐴𝑐𝑐 𝑀 (𝑥 ∷ 𝐾)
  • 31. Multiset Ordering• 4. Case Analysis• By definition, 𝐴𝑐𝑐 𝑀 (𝑥 ∷ 𝐿) is equivalent to ∀𝐾, 𝐾 < 𝑀 (𝑥 ∷ 𝐿) ⇒ 𝐴𝑐𝑐 𝑀 (𝐾)• And there are 3 patterns: 1. 𝐾 includes 𝑥 2. 𝐾 includes 𝑦s s.t. 𝑦 < 𝑥, and 𝐾 minus all such 𝑦 is equal to 𝐿 3. 𝐾 includes 𝑦s s.t. 𝑦 < 𝑥, and 𝐾 minus all such 𝑦 is less than 𝐿• Each pattern is proved using the Inductive Hypotheses.
  • 32. Decidability• Now, decidability can be proved by induction on the size of sequent.
  • 33. Implementation Detail•
  • 34. IPC Proposition (Coq) Inductive PProp:Set :=• | PPbot : PProp | PPatom : nat -> PProp | PPimpl : PProp -> PProp -> PProp | PPconj : PProp -> PProp -> PProp | PPdisj : PProp -> PProp -> PProp.
  • 35. Cut-free LJ (Coq) Inductive LJ_provable : list PProp -> PProp -> Prop :=• | LJ_perm P1 L1 L2 : Permutation L1 L2 -> LJ_provable L1 P1 -> LJ_provable L2 P1 | LJ_weak P1 P2 L1 : LJ_provable L1 P2 -> LJ_provable (P1::L1) P2 | LJ_contr P1 P2 L1 : LJ_provable (P1::P1::L1) P2 -> LJ_provable (P1::L1) P2 …
  • 36. Exchange rule• Exchange rule : Γ, 𝐴, 𝐵, Δ ⊢ 𝐺 𝑒𝑥𝑐ℎ Γ, 𝐵, 𝐴, Δ ⊢ 𝐺 is replaced by more useful Γ⊢ 𝐺 ′ ⊢ 𝐺 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛 Γ where Γ, Γ′ are permutation
  • 37. Permutation Compatibility (Coq)Instance LJ_provable_compat : Proper (@Permutation _==>eq==>iff) LJ_provable.• Allows rewriting over Permutation equality
  • 38. Permutation solver (Coq)• Permutation should be solved automaticallyLtac perm := match goal with…
  • 39. Further implementation plan•
  • 40. Further implementation plan• Refactoring (1) : improve Permutation- associated tactics – A smarter auto-unifying tactics is needed – Write tactics using Objective Caml• Refactoring (2) : use Ssreflect tacticals – This makes the proof more manageable
  • 41. Further implementation plan• Refactoring (3) : change proof order – Contraction first, cut next – It will make the proof shorter• Refactoring (4) : discard Multiset Ordering – If we choose appropriate weight function of Propositional Formula, we don’t need Multiset Ordering. (See [Hudelmaier]) – It also enables us to analyze complexity of this procedure
  • 42. Further implementation plan• Refactoring (5) : Proof of completeness – Now completeness theorem depends on the decidability• New Theorem (1) : Other Syntaxes – NJ and HJ may be introduced• New Theorem (2) : Other Semantics – Heyting Algebra
  • 43. Further implementation plan• New Theorem (3) : Other decision procedure – Decision procedure using semantics (if any) – More efficient decision procedure (especially 𝑂(𝑁 log 𝑁)-space decision procedure)• New Theorem (4) : Complexity – Proof of PSPACE-completeness
  • 44. Source code• Source codes are:• https://github.com/qnighy/IPC-Coq
  • 45. おわり1. Task & Known results2. Brief methodology of the proof 1. Cut elimination 2. Contraction elimination 3. → 𝐿 elimination 4. Proof of strictly-decreasingness3. Implementation detail4. Further implementation plan
  • 46. References• [Dyckhoff] Roy Dyckhoff, Contraction-free Sequent Calculi for Intuitionistic Logic, The Journal of Symbolic Logic, Vol. 57, No.3, 1992, pp. 795 – 807• [Statman] Richard Statman, Intuitionistic Propositional Logic is Polynomial-Space Complete, Theoretical Computer Science 9, 1979, pp. 67 – 72• [Hudelmaier] Jörg Hudelmaier, An O(n log n)-Space Decision Procedure for Intuitionistic Propositional Logic, Journal of Logic and Computation, Vol. 3, Issue 1, pp. 63-75