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

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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

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

Extracting Higher-Order Goals from the Mizar Mathematical Library

Chad E. Brown, Josef Urban

Czech Technical University in Prague

July 2016

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

Outline

Introduction Three Constructs Simple Type Theory Idealized Mizar Translation Results Conclusion

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

Introduction

◮ Extension of Urban’s MPTP translation of Mizar’s

MML from FO to HO

◮ Constructs fitting HO:

◮ Schemes ◮ Global Choice Operator ◮ Fraenkel Terms

◮ Provides problems for HO ATPs ◮ Evaluation of Satallax and LEO-II on some of these

problems.

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

Outline

Introduction Three Constructs Simple Type Theory Idealized Mizar Translation Results Conclusion

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

Schemes

∀A.∀P. (∀x, y, z.Pxy ∧ Pxz → y = z) → ∃X.∀x.x ∈ X ↔ ∃y.y ∈ A ∧ Pyx

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

Schemes

∀A.∀P. (∀x, y, z.Pxy ∧ Pxz → y = z) → ∃X.∀x.x ∈ X ↔ ∃y.y ∈ A ∧ Pyx

◮ Problem: Can’t write ∀P in FO

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

Schemes

∀A.∀P. (∀x, y, z.Pxy ∧ Pxz → y = z) → ∃X.∀x.x ∈ X ↔ ∃y.y ∈ A ∧ Pyx

◮ Problem: Can’t write ∀P in FO ◮ For proving schemes, not an issue.

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

Schemes

∀A.∀P. (∀x, y, z.Pxy ∧ Pxz → y = z) → ∃X.∀x.x ∈ X ↔ ∃y.y ∈ A ∧ Pyx

◮ Problem: Can’t write ∀P in FO ◮ For proving schemes, not an issue. ◮ The issue with using schemes.

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

Using Schemes

can be used to prove Proof:

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

Using Schemes

can be used to prove Proof:

◮ Let A and P be given.

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

Using Schemes

can be used to prove Proof:

◮ Let A and P be given. ◮ Instantiate the A and P from Replacement with

A := A P[x, y] := x = y ∧ P[x]

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

MPTP and Schemes

◮ How does MPTP handle schemes? ◮ Solution staying in FO: Give some FO instances of the

scheme

◮ HO Solution: Simply quantify over P ◮ HO ATP must find the instance as part of the search,

e.g., λxy.x = y ∧ Px

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

MPTP and Schemes

◮ How does MPTP handle schemes? ◮ Solution staying in FO: Give some FO instances of the

scheme

◮ HO Solution: Simply quantify over P ◮ HO ATP must find the instance as part of the search,

e.g., λxy.x = y ∧ Px

◮ Finding these instantiations is nontrivial.

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

MPTP and Schemes

◮ How does MPTP handle schemes? ◮ Solution staying in FO: Give some FO instances of the

scheme

◮ HO Solution: Simply quantify over P ◮ HO ATP must find the instance as part of the search,

e.g., λxy.x = y ∧ Px

◮ Finding these instantiations is nontrivial. ◮ Redundancy. These also work:

λxy.x = y ∧ Py λxy.y = x ∧ Px λxy.y = x ∧ Py

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

MPTP and Schemes

◮ How does MPTP handle schemes? ◮ Solution staying in FO: Give some FO instances of the

scheme

◮ HO Solution: Simply quantify over P ◮ HO ATP must find the instance as part of the search,

e.g., λxy.x = y ∧ Px

◮ Finding these instantiations is nontrivial. ◮ Redundancy. These also work:

λxy.x = y ∧ Py λxy.y = x ∧ Px λxy.y = x ∧ Py

◮ Many other instantiations of roughly this size don’t

work.

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

Global Choice

the A where A is a Mizar type. Examples of Mizar types:

◮ set

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

Global Choice

the A where A is a Mizar type. Examples of Mizar types:

◮ set

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

Global Choice

the A where A is a Mizar type. Examples of Mizar types:

◮ set ◮ Element of X whenever X has type set

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

Global Choice

the A where A is a Mizar type. Examples of Mizar types:

◮ set ◮ Element of X whenever X has type set

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

Global Choice

the A where A is a Mizar type.

◮ All Mizar types are nonempty.

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Global Choice

the A where A is a Mizar type.

◮ All Mizar types are nonempty. ◮ Element of X means ∈ X if X is nonempty, and means

= ∅ if X is empty.

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

Global Choice

the A where A is a Mizar type.

◮ All Mizar types are nonempty. ◮ Element of X means ∈ X if X is nonempty, and means

= ∅ if X is empty.

◮ If X is empty the Element of X is ∅ when X is empty.

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

Global Choice

the A where A is a Mizar type.

◮ All Mizar types are nonempty. ◮ Element of X means ∈ X if X is nonempty, and means

= ∅ if X is empty.

◮ If X is empty the Element of X is ∅ when X is empty.

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

Global Choice and MPTP

◮ MPTP in FO?

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Global Choice and MPTP

◮ MPTP in FO? ◮ Deanonymize:

◮ Suppose A is a Mizar type depending on x, y.

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Global Choice and MPTP

◮ MPTP in FO? ◮ Deanonymize:

◮ Suppose A is a Mizar type depending on x, y. ◮ Translate the A as h(x, y) where

∀x, y.φA(h(x, y), x, y) where φA is the FO translation of A.

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

Global Choice and MPTP

◮ MPTP in FO? ◮ Deanonymize:

◮ Suppose A is a Mizar type depending on x, y. ◮ Translate the A as h(x, y) where

∀x, y.φA(h(x, y), x, y) where φA is the FO translation of A.

◮ MPTP in HO? ◮ Translate the A as ΦA (ε(λu.ΦA u x y) x y)

where ε is a choice operator and ΦA is the HO translation of A.

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

Global Choice Example

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

Global Choice Example

FO MPTP translation. Let x ˆ ∈Y be translation of x Element of Y .

◮ ∀x.F(x) = G(x) ◮ ∀x.h(x)ˆ

∈F(x)

◮ ∀x.k(x)ˆ

∈G(x) Conjecture: ∀x.h(x) = k(x)

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

Global Choice Example

FO MPTP translation. Let x ˆ ∈Y be translation of x Element of Y .

◮ ∀x.F(x) = G(x) ◮ ∀x.h(x)ˆ

∈F(x)

◮ ∀x.k(x)ˆ

∈G(x) Conjecture: ∀x.h(x) = k(x) Not actually a theorem!

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

Global Choice Example

FO MPTP translation. Let x ˆ ∈Y be translation of x Element of Y .

◮ ∀x.F(x) = G(x) ◮ ∀x.h(x)ˆ

∈F(x)

◮ ∀x.k(x)ˆ

∈G(x) Conjecture: ∀x.h(x) = k(x) Not actually a theorem! HO MPTP translation. Axioms:

◮ ∀qx.qx → q(εq) (Choice) ◮ ∀x.Fx = Gx

Conjecture: ∀x.ε(λu.uˆ ∈(Fx)) = ε(λu.uˆ ∈(Gx)) Easy theorem.

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

Fraenkel Terms

{t where x is A : P} where

◮ t is a Mizar term, ◮ A is a “setlike” Mizar type and ◮ P is a Mizar proposition.

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

Fraenkel Terms

{t where x is A : P} where

◮ t is a Mizar term, ◮ A is a “setlike” Mizar type and ◮ P is a Mizar proposition.

{t where x1 is A1, . . . , xn is An : P}

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

Fraenkel Example

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

Fraenkel Example

FO MPTP version, deanonymize:

◮ h(Y ) such that

∀z.z ∈ h(Y ) ↔ ∃x.x ˆ ∈Y ∧ z = {x} ∧ P(x)

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

Fraenkel Example

FO MPTP version, deanonymize:

◮ h(Y ) such that

∀z.z ∈ h(Y ) ↔ ∃x.x ˆ ∈Y ∧ z = {x} ∧ P(x)

◮ k(Y ) such that

∀z.z ∈ k(Y ) ↔ ∃x.x ˆ ∈Y ∧ z = {x} ∧ Q(x)

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

Fraenkel Example

FO MPTP version, deanonymize:

◮ h(Y ) such that

∀z.z ∈ h(Y ) ↔ ∃x.x ˆ ∈Y ∧ z = {x} ∧ P(x)

◮ k(Y ) such that

∀z.z ∈ k(Y ) ↔ ∃x.x ˆ ∈Y ∧ z = {x} ∧ Q(x)

◮ Assume ∀x.P(x) ↔ Q(x).

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

Fraenkel Example

FO MPTP version, deanonymize:

◮ h(Y ) such that

∀z.z ∈ h(Y ) ↔ ∃x.x ˆ ∈Y ∧ z = {x} ∧ P(x)

◮ k(Y ) such that

∀z.z ∈ k(Y ) ↔ ∃x.x ˆ ∈Y ∧ z = {x} ∧ Q(x)

◮ Assume ∀x.P(x) ↔ Q(x). ◮ Prove ∀Y .h(Y ) = k(Y )

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

Fraenkel Example

FO MPTP version, deanonymize:

◮ h(Y ) such that

∀z.z ∈ h(Y ) ↔ ∃x.x ˆ ∈Y ∧ z = {x} ∧ P(x)

◮ k(Y ) such that

∀z.z ∈ k(Y ) ↔ ∃x.x ˆ ∈Y ∧ z = {x} ∧ Q(x)

◮ Assume ∀x.P(x) ↔ Q(x). ◮ Prove ∀Y .h(Y ) = k(Y ) ◮ Set extensionality and information about ˆ

∈ is also given, which is enough to make the FO problem provable.

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Fraenkel Example

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Fraenkel Example

HO MPTP version, use a declared constant replSep1

◮ replSep1 : (ιo)(ιι)(ιo)ι.

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

Fraenkel Example

HO MPTP version, use a declared constant replSep1

◮ replSep1 : (ιo)(ιι)(ιo)ι. ◮ Assume ∀x.Px ↔ Qx.

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

Fraenkel Example

HO MPTP version, use a declared constant replSep1

◮ replSep1 : (ιo)(ιι)(ιo)ι. ◮ Assume ∀x.Px ↔ Qx. ◮ Conjecture:

replSep1 (λx.x ˆ ∈Y ) (λx.{x}) (λx.Px) = replSep1 (λx.x ˆ ∈Y ) (λx.{x}) (λx.Qx)

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

Fraenkel Example

HO MPTP version, use a declared constant replSep1

◮ replSep1 : (ιo)(ιι)(ιo)ι. ◮ Assume ∀x.Px ↔ Qx. ◮ Conjecture:

replSep1 (λx.x ˆ ∈Y ) (λx.{x}) (λx.Px) = replSep1 (λx.x ˆ ∈Y ) (λx.{x}) (λx.Qx)

◮ Easy HO theorem (using extensionality rules).

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

Outline

Introduction Three Constructs Simple Type Theory Idealized Mizar Translation Results Conclusion

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

Simple Type Theory

Simple types:

◮ ι - individuals (Mizar sets/objects) ◮ o - propositions ◮ αβ - functions from α to β

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

Simple Type Theory

Simple types:

◮ ι - individuals (Mizar sets/objects) ◮ o - propositions ◮ αβ - functions from α to β

Simply typed terms:

◮ Typed variables: x : α ◮ Typed constants: c : α ◮ Application: st : β where s : αβ and t : α ◮ Abstraction: λx.t : αβ where x : α and t : β ◮ Implication: s → t : o where s, t : o ◮ Quantification: ∀x.t : o where x : α and t : o

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

Simple Type Theory

Simple types:

◮ ι - individuals (Mizar sets/objects) ◮ o - propositions ◮ αβ - functions from α to β

Simply typed terms:

◮ Typed variables: x : α ◮ Typed constants: c : α ◮ Application: st : β where s : αβ and t : α ◮ Abstraction: λx.t : αβ where x : α and t : β ◮ Implication: s → t : o where s, t : o ◮ Quantification: ∀x.t : o where x : α and t : o

Proofs: Usual rules, including extensionality

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

Outline

Introduction Three Constructs Simple Type Theory Idealized Mizar Translation Results Conclusion

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

Variables and Constants

Inherit variables and constants from STT.

◮ x : ι object variables, c : ι object constants

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

Variables and Constants

Inherit variables and constants from STT.

◮ x : ι object variables, c : ι object constants ◮ F : ι . . . ι n

ι function variables of arity n

◮ f : ι . . . ι n

ι function constants of arity n

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Variables and Constants

Inherit variables and constants from STT.

◮ x : ι object variables, c : ι object constants ◮ F : ι . . . ι n

ι function variables of arity n

◮ f : ι . . . ι n

ι function constants of arity n

◮ P : ι . . . ι n

  • predicate variables of arity n

◮ p : ι . . . ι n

  • predicate constants of arity n
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Extracting Higher-Order Goals from the Mizar Mathematical Library Brown, Urban Introduction Three Constructs Simple Type Theory Idealized Mizar Translation Results Conclusion

Variables and Constants

Inherit variables and constants from STT.

◮ x : ι object variables, c : ι object constants ◮ F : ι . . . ι n

ι function variables of arity n

◮ f : ι . . . ι n

ι function constants of arity n

◮ P : ι . . . ι n

  • predicate variables of arity n

◮ p : ι . . . ι n

  • predicate constants of arity n

◮ Predicate constants also play the role of Mizar modes or

attributes.

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

M-types, M-terms and M-propositions

Mutually recursive definitions of

◮ M-types A, B, . . . ◮ M-terms S, T, . . . ◮ M-propositions Φ, Ψ, . . .

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

M-types, M-terms and M-propositions

Mutually recursive definitions of

◮ M-types A, B, . . . ◮ M-terms S, T, . . . ◮ M-propositions Φ, Ψ, . . .

M-types:

◮ set ◮ p(·, T1, . . . , Tn) where p n + 1-ary predicate constant (p

mode)

◮ q A and non q A where q is a unary predicate constant

(q attribute)

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M-types, M-terms and M-propositions

Mutually recursive definitions of

◮ M-types A, B, . . . ◮ M-terms S, T, . . . ◮ M-propositions Φ, Ψ, . . .

M-terms:

◮ object variables and object constants ◮ F(T1, . . . , Tn) ◮ f (T1, . . . , Tn)

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M-types, M-terms and M-propositions

Mutually recursive definitions of

◮ M-types A, B, . . . ◮ M-terms S, T, . . . ◮ M-propositions Φ, Ψ, . . .

M-terms:

◮ object variables and object constants ◮ F(T1, . . . , Tn) ◮ f (T1, . . . , Tn) ◮ (the A) (global choice)

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

M-types, M-terms and M-propositions

Mutually recursive definitions of

◮ M-types A, B, . . . ◮ M-terms S, T, . . . ◮ M-propositions Φ, Ψ, . . .

M-terms:

◮ object variables and object constants ◮ F(T1, . . . , Tn) ◮ f (T1, . . . , Tn) ◮ (the A) (global choice) ◮ {T where x1 is A1, . . . xn is An : Φ} (Fraenkel terms)

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

M-types, M-terms and M-propositions

Mutually recursive definitions of

◮ M-types A, B, . . . ◮ M-terms S, T, . . . ◮ M-propositions Φ, Ψ, . . .

M-propositions

◮ P[T1, . . . , Tn] ◮ p[T1, . . . , Tn] ◮ (S = T) and (S in T) ◮ (not Φ) ◮ (Φ & Ψ), (Φ or Ψ), (Φ implies Ψ) and (Φ iff Ψ) ◮ (for x being A holds Φ) ◮ (ex x being A st Φ)

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M-statements

A prefix Γ is a list of variable declarations:

◮ x : A ◮ F(A1, . . . , An) : B ◮ P[A1, . . . , An]

An M-statement is a prefix and an M-proposition.

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M-statements

A prefix Γ is a list of variable declarations:

◮ x : A ◮ F(A1, . . . , An) : B ◮ P[A1, . . . , An]

An M-statement is a prefix and an M-proposition. Example:

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

M-statements

A prefix Γ is a list of variable declarations:

◮ x : A ◮ F(A1, . . . , An) : B ◮ P[A1, . . . , An]

An M-statement is a prefix and an M-proposition. Example:

◮ Γ is the prefix

A : set, P[set]

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

M-statements

A prefix Γ is a list of variable declarations:

◮ x : A ◮ F(A1, . . . , An) : B ◮ P[A1, . . . , An]

An M-statement is a prefix and an M-proposition. Example:

◮ Γ is the prefix

A : set, P[set]

◮ Φ is the M-proposition

ex X being set st for x being set holds x in X iff x in A & P(x)

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

M-statements

A prefix Γ is a list of variable declarations:

◮ x : A ◮ F(A1, . . . , An) : B ◮ P[A1, . . . , An]

An M-statement is a prefix and an M-proposition. Example:

◮ Γ is the prefix

A : set, P[set]

◮ Φ is the M-proposition

ex X being set st for x being set holds x in X iff x in A & P(x)

◮ The M-statement is (Γ, Φ)

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

Outline

Introduction Three Constructs Simple Type Theory Idealized Mizar Translation Results Conclusion

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

Translation

◮ M-types A translate to terms A of type ιo ◮ M-terms T translate to terms T of type ι ◮ M-propositions Φ translate to terms Φ of type o

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

Translation

◮ M-types A translate to terms A of type ιo ◮ M-terms T translate to terms T of type ι

(choice, Fraenkels)

◮ M-propositions Φ translate to terms Φ of type o

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

Translation

◮ M-types A translate to terms A of type ιo ◮ M-terms T translate to terms T of type ι

(choice, Fraenkels)

◮ M-propositions Φ translate to terms Φ of type o ◮ M-statements (Γ, Φ) also translate to terms (Γ, Φ) of

type o (schemes)

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

Translation

◮ M-types A translate to terms A of type ιo ◮ M-terms T translate to terms T of type ι

(choice, Fraenkels)

◮ M-propositions Φ translate to terms Φ of type o ◮ M-statements (Γ, Φ) also translate to terms (Γ, Φ) of

type o (schemes)

◮ the A = εA

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

Translation

◮ M-types A translate to terms A of type ιo ◮ M-terms T translate to terms T of type ι

(choice, Fraenkels)

◮ M-propositions Φ translate to terms Φ of type o ◮ M-statements (Γ, Φ) also translate to terms (Γ, Φ) of

type o (schemes)

◮ the A = εA ◮ {T where x is A : Φ} =

replSep1 A (λx.T) (λx.Φ)

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

Translation

◮ M-types A translate to terms A of type ιo ◮ M-terms T translate to terms T of type ι

(choice, Fraenkels)

◮ M-propositions Φ translate to terms Φ of type o ◮ M-statements (Γ, Φ) also translate to terms (Γ, Φ) of

type o (schemes)

◮ the A = εA ◮ {T where x is A : Φ} =

replSep1 A (λx.T) (λx.Φ)

◮ (·, Φ) = Φ. ◮ ((x : A, Γ), Φ) = ∀x.A x → (Γ, Φ).

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

Translation

◮ M-types A translate to terms A of type ιo ◮ M-terms T translate to terms T of type ι

(choice, Fraenkels)

◮ M-propositions Φ translate to terms Φ of type o ◮ M-statements (Γ, Φ) also translate to terms (Γ, Φ) of

type o (schemes)

◮ the A = εA ◮ {T where x is A : Φ} =

replSep1 A (λx.T) (λx.Φ)

◮ (·, Φ) = Φ. ◮ ((x : A, Γ), Φ) = ∀x.A x → (Γ, Φ). ◮ ((F(A1, . . . , An) : B, Γ), Φ) = ∀F.(∀x1.A1 x1 →

. . . → ∀xn.An xn → B (Fx1 · · · xn)) → (Γ, Φ).

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

Translation

◮ M-types A translate to terms A of type ιo ◮ M-terms T translate to terms T of type ι

(choice, Fraenkels)

◮ M-propositions Φ translate to terms Φ of type o ◮ M-statements (Γ, Φ) also translate to terms (Γ, Φ) of

type o (schemes)

◮ the A = εA ◮ {T where x is A : Φ} =

replSep1 A (λx.T) (λx.Φ)

◮ (·, Φ) = Φ. ◮ ((x : A, Γ), Φ) = ∀x.A x → (Γ, Φ). ◮ ((F(A1, . . . , An) : B, Γ), Φ) = ∀F.(∀x1.A1 x1 →

. . . → ∀xn.An xn → B (Fx1 · · · xn)) → (Γ, Φ).

◮ ((P[A1, . . . , An], Γ), Φ) = ∀P.(Γ, Φ).

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

Translation (General Fraenkel Case)

{T where x1 is A1, . . . xn is An : Φ} = replSepn A1 (λx1.A2) · · · (λx1 · · · xn−1.An) (λx1 · · · xn.T) (λx1 · · · xn.Φ)

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

Translation (General Fraenkel Case)

{T where x1 is A1, . . . xn is An : Φ} = replSepn A1 (λx1.A2) · · · (λx1 · · · xn−1.An) (λx1 · · · xn.T) (λx1 · · · xn.Φ)

◮ replSepn : (ιo) · · · (ι · · · ιo)(ι · · · ιι)(ι · · · ιo)

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

Translation (General Fraenkel Case)

{T where x1 is A1, . . . xn is An : Φ} = replSepn A1 (λx1.A2) · · · (λx1 · · · xn−1.An) (λx1 · · · xn.T) (λx1 · · · xn.Φ)

◮ replSepn : (ιo) · · · (ι · · · ιo)(ι · · · ιι)(ι · · · ιo) ◮ Axioms for replSepn:

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

Translation (General Fraenkel Case)

{T where x1 is A1, . . . xn is An : Φ} = replSepn A1 (λx1.A2) · · · (λx1 · · · xn−1.An) (λx1 · · · xn.T) (λx1 · · · xn.Φ)

◮ replSepn : (ιo) · · · (ι · · · ιo)(ι · · · ιι)(ι · · · ιo) ◮ Axioms for replSepn: ◮ replSepEn: If z ∈ replSepnA1 · · · AnFP, then there

exist x1, . . . , xn such that A1x1, . . ., Anx1 · · · xn, z = Fx1 · · · xn and Px1 · · · xn.

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

Translation (General Fraenkel Case)

{T where x1 is A1, . . . xn is An : Φ} = replSepn A1 (λx1.A2) · · · (λx1 · · · xn−1.An) (λx1 · · · xn.T) (λx1 · · · xn.Φ)

◮ replSepn : (ιo) · · · (ι · · · ιo)(ι · · · ιι)(ι · · · ιo) ◮ Axioms for replSepn: ◮ replSepEn: If z ∈ replSepnA1 · · · AnFP, then there

exist x1, . . . , xn such that A1x1, . . ., Anx1 · · · xn, z = Fx1 · · · xn and Px1 · · · xn.

◮ replSepIn: If A1 is “setlike”, A1x1, . . ., Anx1 · · · xn−1

is “setlike”, Anx1 · · · xn and Px1 · · · xn, then Fx1 · · · xn ∈ replSepnA1 · · · AnFP.

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

Outline

Introduction Three Constructs Simple Type Theory Idealized Mizar Translation Results Conclusion

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

Problems Sets

◮ Top level justifications (by)

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

Problems Sets

◮ Top level justifications (by)

◮ involving global choice: 47

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

Problems Sets

◮ Top level justifications (by)

◮ involving global choice: 47 ◮ involving Fraenkels: 245

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

Problems Sets

◮ Top level justifications (by)

◮ involving global choice: 47 ◮ involving Fraenkels: 245

◮ Scheme justifications (from): 10192

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

Problems Sets

◮ Top level justifications (by)

◮ involving global choice: 47 ◮ involving Fraenkels: 245

◮ Scheme justifications (from): 10192 ◮ Scheme Proofs: 610

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

Global Choice Justification Problems

Ran Satallax and LEO-II for 5 minutes with default settings.

◮ 47 total ◮ Satallax: 24 (51%) ◮ LEO-II: 28 (60%) ◮ Either: 30 (64%)

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

Fraenkel Justification Problems

Ran Satallax and LEO-II for 5 minutes with default settings.

◮ 245 total ◮ Satallax: 126 (52%) ◮ LEO-II: 88 (36%) ◮ Either: 165 (67%)

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

Fraenkel Justification Problems

Ran Satallax and LEO-II for 5 minutes with default settings.

◮ 245 total ◮ Satallax: 126 (52%) ◮ LEO-II: 88 (36%) ◮ Either: 165 (67%)

Idea: Use E to indicate which FO axioms it needs to do the FO version. “Prune” the HO version.

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

Fraenkel Justification Problems

Ran Satallax and LEO-II for 5 minutes with default settings.

◮ 245 total ◮ Satallax: 126 (52%) ◮ LEO-II: 88 (36%) ◮ Either: 165 (67%)

Idea: Use E to indicate which FO axioms it needs to do the FO version. “Prune” the HO version.

◮ 245 total ◮ Satallax: 159 (65%) ◮ LEO-II: 155 (63%) ◮ Either: 192 (78%)

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

Scheme Justification Problems

Ran Satallax and LEO-II for 5 minutes with default settings.

◮ 10192 total ◮ Satallax: 5608 (55%) ◮ LEO-II: 1524 (15%) ◮ Either: 6072 (60%)

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

Full Schemes

Ran Satallax and LEO-II for 5 minutes with default settings.

◮ 610 total ◮ Satallax: 31 (5%) ◮ LEO-II: 67 (11%) ◮ Either: 81 (13%)

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

Outline

Introduction Three Constructs Simple Type Theory Idealized Mizar Translation Results Conclusion

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

Conclusion

◮ Some Mizar constructs translate in a “higher-order”

way.

◮ The translation gives higher-order problem sets

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

Conclusion

◮ Some Mizar constructs translate in a “higher-order”

way.

◮ The translation gives higher-order problem sets ◮ ...which are challenging for HO ATPs.

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

Conclusion

◮ Some Mizar constructs translate in a “higher-order”

way.

◮ The translation gives higher-order problem sets ◮ ...which are challenging for HO ATPs. ◮ Future Work: ◮ Should theorem provers always prove “Mizar-obvious”

problems?

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

Conclusion

◮ Some Mizar constructs translate in a “higher-order”

way.

◮ The translation gives higher-order problem sets ◮ ...which are challenging for HO ATPs. ◮ Future Work: ◮ Should theorem provers always prove “Mizar-obvious”

problems?

◮ Should Mizar’s type system be preserved by the

translation and used by theorem provers?

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

Conclusion

◮ Some Mizar constructs translate in a “higher-order”

way.

◮ The translation gives higher-order problem sets ◮ ...which are challenging for HO ATPs. ◮ Future Work: ◮ Should theorem provers always prove “Mizar-obvious”

problems?

◮ Should Mizar’s type system be preserved by the

translation and used by theorem provers?

◮ Translating proofs back to Mizar