Extracting Higher-Order Goals from the Introduction Three - PowerPoint PPT Presentation

Extracting Higher-Order Goals from the Mizar Mathematical Library Brown, Urban Extracting Higher-Order Goals from the Introduction Three Constructs Mizar Mathematical Library Simple Type Theory Idealized Mizar Chad E. Brown, Josef Urban Translation Results Czech Technical University in Prague Conclusion July 2016

Extracting Outline Higher-Order Goals from the Mizar Mathematical Library Introduction Brown, Urban Introduction Three Constructs Three Constructs Simple Type Theory Simple Type Theory Idealized Mizar Translation Idealized Mizar Results Conclusion Translation Results Conclusion

Extracting Introduction Higher-Order Goals from the Mizar Mathematical Library Brown, Urban ◮ Extension of Urban’s MPTP translation of Mizar’s Introduction Three Constructs MML from FO to HO Simple Type ◮ Constructs fitting HO: Theory ◮ Schemes Idealized Mizar ◮ Global Choice Operator Translation ◮ Fraenkel Terms Results ◮ Provides problems for HO ATPs Conclusion ◮ Evaluation of Satallax and LEO-II on some of these problems.

Extracting Outline Higher-Order Goals from the Mizar Mathematical Library Introduction Brown, Urban Introduction Three Constructs Three Constructs Simple Type Theory Simple Type Theory Idealized Mizar Translation Idealized Mizar Results Conclusion Translation Results Conclusion

Extracting Schemes Higher-Order Goals from the Mizar Mathematical Library Brown, Urban Introduction Three Constructs Simple Type Theory ∀ A . ∀ P . Idealized Mizar ( ∀ x , y , z . Pxy ∧ Pxz → y = z ) → Translation ∃ X . ∀ x . x ∈ X ↔ ∃ y . y ∈ A ∧ Pyx Results Conclusion

Extracting Schemes Higher-Order Goals from the Mizar Mathematical Library Brown, Urban Introduction Three Constructs Simple Type Theory ∀ A . ∀ P . Idealized Mizar ( ∀ x , y , z . Pxy ∧ Pxz → y = z ) → Translation ∃ X . ∀ x . x ∈ X ↔ ∃ y . y ∈ A ∧ Pyx Results Conclusion ◮ Problem: Can’t write ∀ P in FO

Extracting Schemes Higher-Order Goals from the Mizar Mathematical Library Brown, Urban Introduction Three Constructs Simple Type Theory ∀ A . ∀ P . Idealized Mizar ( ∀ x , y , z . Pxy ∧ Pxz → y = z ) → Translation ∃ X . ∀ x . x ∈ X ↔ ∃ y . y ∈ A ∧ Pyx Results Conclusion ◮ Problem: Can’t write ∀ P in FO ◮ For proving schemes, not an issue.

Extracting Schemes Higher-Order Goals from the Mizar Mathematical Library Brown, Urban Introduction Three Constructs Simple Type Theory ∀ A . ∀ P . Idealized Mizar ( ∀ x , y , z . Pxy ∧ Pxz → y = z ) → Translation ∃ X . ∀ x . x ∈ X ↔ ∃ y . y ∈ A ∧ Pyx Results Conclusion ◮ Problem: Can’t write ∀ P in FO ◮ For proving schemes, not an issue. ◮ The issue with using schemes.

Extracting Using Schemes Higher-Order Goals from the Mizar Mathematical Library Brown, Urban Introduction Three Constructs can be used to prove Simple Type Theory Idealized Mizar Translation Proof: Results Conclusion

Extracting Using Schemes Higher-Order Goals from the Mizar Mathematical Library Brown, Urban Introduction Three Constructs can be used to prove Simple Type Theory Idealized Mizar Translation Proof: Results ◮ Let A and P be given. Conclusion

Extracting Using Schemes Higher-Order Goals from the Mizar Mathematical Library Brown, Urban Introduction Three Constructs can be used to prove Simple Type Theory Idealized Mizar Translation Proof: Results ◮ Let A and P be given. Conclusion ◮ Instantiate the A and P from Replacement with A := A P [ x , y ] := x = y ∧ P [ x ]

Extracting MPTP and Schemes Higher-Order Goals from the Mizar ◮ How does MPTP handle schemes? Mathematical Library ◮ Solution staying in FO: Give some FO instances of the Brown, Urban scheme Introduction ◮ HO Solution: Simply quantify over P Three Constructs ◮ HO ATP must find the instance as part of the search, Simple Type e.g., Theory Idealized Mizar λ xy . x = y ∧ Px Translation Results Conclusion

Extracting MPTP and Schemes Higher-Order Goals from the Mizar ◮ How does MPTP handle schemes? Mathematical Library ◮ Solution staying in FO: Give some FO instances of the Brown, Urban scheme Introduction ◮ HO Solution: Simply quantify over P Three Constructs ◮ HO ATP must find the instance as part of the search, Simple Type e.g., Theory Idealized Mizar λ xy . x = y ∧ Px Translation ◮ Finding these instantiations is nontrivial. Results Conclusion

Extracting MPTP and Schemes Higher-Order Goals from the Mizar ◮ How does MPTP handle schemes? Mathematical Library ◮ Solution staying in FO: Give some FO instances of the Brown, Urban scheme Introduction ◮ HO Solution: Simply quantify over P Three Constructs ◮ HO ATP must find the instance as part of the search, Simple Type e.g., Theory Idealized Mizar λ xy . x = y ∧ Px Translation ◮ Finding these instantiations is nontrivial. Results ◮ Redundancy. These also work: Conclusion λ xy . x = y ∧ Py λ xy . y = x ∧ Px λ xy . y = x ∧ Py

Extracting MPTP and Schemes Higher-Order Goals from the Mizar ◮ How does MPTP handle schemes? Mathematical Library ◮ Solution staying in FO: Give some FO instances of the Brown, Urban scheme Introduction ◮ HO Solution: Simply quantify over P Three Constructs ◮ HO ATP must find the instance as part of the search, Simple Type e.g., Theory Idealized Mizar λ xy . x = y ∧ Px Translation ◮ Finding these instantiations is nontrivial. Results ◮ Redundancy. These also work: Conclusion λ xy . x = y ∧ Py λ xy . y = x ∧ Px λ xy . y = x ∧ Py ◮ Many other instantiations of roughly this size don’t work.

Extracting Global Choice Higher-Order Goals from the Mizar Mathematical Library the A Brown, Urban where A is a Mizar type. Introduction Examples of Mizar types: Three Constructs Simple Type ◮ set Theory Idealized Mizar Translation Results Conclusion

Extracting Global Choice Higher-Order Goals from the Mizar Mathematical Library the A Brown, Urban where A is a Mizar type. Introduction Examples of Mizar types: Three Constructs Simple Type ◮ set Theory Idealized Mizar Translation Results Conclusion

Extracting Global Choice Higher-Order Goals from the Mizar Mathematical Library the A Brown, Urban where A is a Mizar type. Introduction Examples of Mizar types: Three Constructs Simple Type ◮ set Theory Idealized Mizar Translation Results Conclusion ◮ Element of X whenever X has type set

Extracting Global Choice Higher-Order Goals from the Mizar Mathematical Library the A Brown, Urban where A is a Mizar type. Introduction Examples of Mizar types: Three Constructs Simple Type ◮ set Theory Idealized Mizar Translation Results Conclusion ◮ Element of X whenever X has type set

Extracting Global Choice Higher-Order Goals from the Mizar Mathematical Library Brown, Urban the A Introduction Three Constructs where A is a Mizar type. Simple Type Theory ◮ All Mizar types are nonempty. Idealized Mizar Translation Results Conclusion

Extracting Global Choice Higher-Order Goals from the Mizar Mathematical Library Brown, Urban the A Introduction Three Constructs where A is a Mizar type. Simple Type Theory ◮ All Mizar types are nonempty. Idealized Mizar ◮ Element of X means ∈ X if X is nonempty, and means Translation = ∅ if X is empty. Results Conclusion

Extracting Global Choice Higher-Order Goals from the Mizar Mathematical Library Brown, Urban the A Introduction Three Constructs where A is a Mizar type. Simple Type Theory ◮ All Mizar types are nonempty. Idealized Mizar ◮ Element of X means ∈ X if X is nonempty, and means Translation = ∅ if X is empty. Results ◮ If X is empty the Element of X is ∅ when X is empty. Conclusion

Extracting Global Choice Higher-Order Goals from the Mizar Mathematical Library Brown, Urban the A Introduction Three Constructs where A is a Mizar type. Simple Type Theory ◮ All Mizar types are nonempty. Idealized Mizar ◮ Element of X means ∈ X if X is nonempty, and means Translation = ∅ if X is empty. Results ◮ If X is empty the Element of X is ∅ when X is empty. Conclusion

Extracting Global Choice and MPTP Higher-Order Goals from the Mizar Mathematical Library Brown, Urban ◮ MPTP in FO? Introduction Three Constructs Simple Type Theory Idealized Mizar Translation Results Conclusion

Extracting Global Choice and MPTP Higher-Order Goals from the Mizar Mathematical Library Brown, Urban ◮ MPTP in FO? Introduction ◮ Deanonymize: Three Constructs ◮ Suppose A is a Mizar type depending on x , y . Simple Type Theory Idealized Mizar Translation Results Conclusion

Extracting Global Choice and MPTP Higher-Order Goals from the Mizar Mathematical Library Brown, Urban ◮ MPTP in FO? Introduction ◮ Deanonymize: Three Constructs ◮ Suppose A is a Mizar type depending on x , y . Simple Type ◮ Translate the A as h ( x , y ) where Theory Idealized Mizar ∀ x , y .φ A ( h ( x , y ) , x , y ) Translation Results where φ A is the FO translation of A . Conclusion