Flexiformality in Proofs: Bridging between Formal and Informal - - PowerPoint PPT Presentation

Flexiformality in Proofs: Bridging between Formal and Informal Proofs

Michael Kohlhase

Professur für Wissensrepräsentation und -verarbeitung Informatik, FAU Erlangen-Nürnberg http://kwarc.info

• 20. October 2016, Dagstuhl Seminar on “Universality of Proofs”

Kohlhase: Flexiformality in Proofs 1 Dagstuhl

Instead of an Intro: Disclosure of my Angle Today

◮ I want to build a universal digital library of mathematics (in theory graph form) ◮ This is a critical public resource for math system interoperability

◮ Catherine told us about a graph of species in Focalize for Coq/Dedukti

interoperability

◮ Florian did the same for more theorem provers in OMDoc/MMT [KR16] ◮ In the OpenDreamKit project we use the same idea for CAS interop [DIK+16]

◮ The International Mathematical Union (IMU) has chartered a group to prepare a

semantic “Global Digital Maths Library” based on [DLC+14].

◮ Without proofs, all of this will be for nothing,

◮ proofs certify the validity of theorems ◮ modularity/reuse via theory morphisms carry proof obligations ◮ all the aspects that Bill has talked about apply

(thanks for the intro)

Kohlhase: Flexiformality in Proofs 2 Dagstuhl

1 The Flexiformalist Program (inspired by Hilbert)

Kohlhase: Flexiformality in Proofs 3 Dagstuhl

Migration by Stepwise Formalization

◮ Full Formalization is hard

(we have to commit, make explicit)

◮ Let’s look at documents and document collections.

formality number

Kohlhase: Flexiformality in Proofs 4 Dagstuhl

Migration by Stepwise Formalization

◮ Full Formalization is hard

(we have to commit, make explicit)

◮ Let’s look at documents and document collections. ◮ Partial formalization allows us to

◮ formalize stepwise, and ◮ be flexible about the depth of formalization.

formality number

Kohlhase: Flexiformality in Proofs 4 Dagstuhl

What is Informal Mathematical Knowledge

◮ Idea: informal knowledge could be formalized

(but isn’t yet!)

◮ Definition 0.1 The meaning of a knowledge item

is the set of all its formalizations

◮ Problem: What is the space of formalizations? ◮ Definition 0.2 The formal space is the set

F := {S, e | S ∈ F, e ∈ L(S)}, where F is the class of formal systems and L(S) is the language of S. (i.e. every formal expression is a point in F)

◮ Different Logics correspond to different bands ◮ The meaning of D is a set I(D) ⊆ F. ◮ D can be formalized in multiple logics

I(D) forms a cross-section of logic-bands.

Kohlhase: Flexiformality in Proofs 5 Dagstuhl

A Formality Ordering on F

◮ Stepwise formalization looks like this:

Logics E x p r e s s i

• n

s F

• r

m a l S p a c e

Document Space

Less Formal More Formal

D D1 D2 D′

2

D3 D′

3

D′′

3 ◮ Definition 0.3 D is more formal than D′ (write D≪D′), iff I(D) ⊂ I(D′). ◮ This partial ordering relation answers the question of “graded formality” or the

nature of “stepwise formalization” raised above.

Kohlhase: Flexiformality in Proofs 6 Dagstuhl

The Flexiformalist Program (Details in [Koh13])

◮ The development of a regime of partially formalizing

◮ mathematical knowledge into a modular ontology of mathematical theories (content

commons), and

◮ mathematical documents by semantic annotations and links into the content

commons (semantic documents),

◮ The establishment of a software infrastructure with

◮ a distributed network of archives that manage the content commons and collections

• f semantic documents,

◮ semantic web services that perform tasks to support current and future mathematic

practices

◮ active document players that present semantic documents to readers and give access

to respective

◮ the re-development of comprehensive part of mathematical knowledge and the

mathematical documents that carries it into a flexiformal digital library of mathematics.

Kohlhase: Flexiformality in Proofs 7 Dagstuhl

Stephen Watt’s understanding of Flexiformality

A person who is flexiformal:

◮ flexible

(contortionist)

◮ formal

(tuxedo)

Kohlhase: Flexiformality in Proofs 8 Dagstuhl

A Flexiformal Theory Graph to Rule them All

PVS Coq · · · Mizar GAP lmfdb Systems PVS Coq · · · Mizar GAP LMFDB Interface Theories MitM Ontology IMPS TWELF MathScheme MMTCS S T EX OHTML Surface Languages

Kohlhase: Flexiformality in Proofs 9 Dagstuhl

2 Flexiformal Theory Graphs axnd Proofs

Kohlhase: Flexiformality in Proofs 10 Dagstuhl

Background: Redesigning OMDoc

◮ OMDoc: Open Math Documents [Koh06, OMD] models,p document &

knowledge structures

level coverage markup

• bjects/phrases

presentation/content/text math MathML, OpenMath, XHTML statements narrative, some declarations OMDoc, XHTML theories domain theory graphs OMDoc documents grouping OMDoc notations OpenMath MathML rewrites OMDoc quiz, code, . . . ad-hoc OMDoc

Kohlhase: Flexiformality in Proofs 11 Dagstuhl

Background: Redesigning OMDoc

◮ OMDoc: Open Math Documents [Koh06, OMD] models,p document &

knowledge structures

level coverage markup

• bjects/phrases

presentation/content/text math MathML, OpenMath, XHTML statements narrative, some declarations OMDoc, XHTML theories domain theory graphs OMDoc documents grouping OMDoc notations OpenMath MathML rewrites OMDoc quiz, code, . . . ad-hoc OMDoc

◮ MMT: Meta Meta Theory [RK13, MMT] redesigns the formal core of OMDoc

and gives it meaning

level coverage markup

• bjects

content math, literals OpenMath+, MMT statements constant decls/defs OMDoc, MMT theories domain/meta* OMDoc, MMT notations

• ne-level pres/parse

OMDoc, MMT rules typing, reconstruction, . . . Scala

Kohlhase: Flexiformality in Proofs 11 Dagstuhl

Background: Redesigning OMDoc

◮ OMDoc: Open Math Documents [Koh06, OMD] models,p document &

knowledge structures

level coverage markup

• bjects/phrases

presentation/content/text math MathML, OpenMath, XHTML statements narrative, some declarations OMDoc, XHTML theories domain theory graphs OMDoc documents grouping OMDoc notations OpenMath MathML rewrites OMDoc quiz, code, . . . ad-hoc OMDoc

◮ MMT: Meta Meta Theory [RK13, MMT] redesigns the formal core of OMDoc

and gives it meaning

level coverage markup

• bjects

content math, literals OpenMath+, MMT statements constant decls/defs OMDoc, MMT theories domain/meta* OMDoc, MMT notations

• ne-level pres/parse

OMDoc, MMT rules typing, reconstruction, . . . Scala

◮ iMMT: flexiformal MMT designs a formal language for the informal/narrative

parts of OMDoc. (under development with Mihnea Iancu (diss))

Kohlhase: Flexiformality in Proofs 11 Dagstuhl

Background: Redesigning OMDoc

◮ OMDoc: Open Math Documents [Koh06, OMD] models,p document &

knowledge structures

level coverage markup

• bjects/phrases

presentation/content/text math MathML, OpenMath, XHTML statements narrative, some declarations OMDoc, XHTML theories domain theory graphs OMDoc documents grouping OMDoc notations OpenMath MathML rewrites OMDoc quiz, code, . . . ad-hoc OMDoc

◮ MMT: Meta Meta Theory [RK13, MMT] redesigns the formal core of OMDoc

and gives it meaning

level coverage markup

• bjects

content math, literals OpenMath+, MMT statements constant decls/defs OMDoc, MMT theories domain/meta* OMDoc, MMT notations

• ne-level pres/parse

OMDoc, MMT rules typing, reconstruction, . . . Scala

◮ iMMT: flexiformal MMT designs a formal language for the informal/narrative

parts of OMDoc. (under development with Mihnea Iancu (diss))

◮ OMDoc2 ˆ

= MMT + iMMT (needs more language design)

Kohlhase: Flexiformality in Proofs 11 Dagstuhl

How to model Flexiformal Mathematics

◮ I hope to have convinced you: that Math is informal:

◮ foundations unspecified

(what a relief)

◮ natural language & presentation formulae

(humans can disambiguate)

◮ context references

(but math is better than the pack)

◮ Problem: How do we deal with that in our “formal” systems? ◮ Proposed Answer: learn from OpenMath/MathML

◮ referential theory of meaning

(by pointing to symbol definitions)

◮ allow opaque content

(presentation/natural language)

◮ parallel markup

(mix formal/informal recursively at any level)

◮ pluralism at all levels

(object/logic/foundation/metalogic)

◮ underspecification of symbol meaning

extend to statement/paragraph and theory/discourse levels (OMDoc)

Kohlhase: Flexiformality in Proofs 12 Dagstuhl

Parallel Markup e.g. in MathML I

◮ Idea: Combine the presentation and content markup and cross-reference

3 (x+2) 3 ( x+2 ) x + 2 @ 3 @ / + x 2

◮ use e.g. for semantic copy and paste. (click on presentation, follow link and copy

content)

Kohlhase: Flexiformality in Proofs 13 Dagstuhl

Parallel Markup e.g. in MathML II

◮ Concrete Realization in MathML: semantics element with presentation as first

child and content in annotation−xml child

<semantics>...</semantics> <annotation−xml>...</annotation−xml> <mfrac id="M">...</mfrac> <mn id="3">3</mn> <mfenced id="f">...</mfenced> <mi id="x">x</mi> <mo id="p">+</mo> <mn id="2">2</mn> <apply href="M">...</apply> <divide/> <ci href="3">3<ci/> <apply href="f">...</apply> <plus href="p"/> <ci href="x">x</ci> <cn href="2">2</cn>

Kohlhase: Flexiformality in Proofs 14 Dagstuhl

Parallel Markup at the Discourse Level

◮ Observation: Parallel markup supports two workflows:

◮ formalization: the annotation of presentation formulae with (multiple) formalizations

and

◮ presentation: the annotation of content formulae with (multiple) presentations.

◮ Idea: At the discourse level, the duality between presentation and content

manifests itself as that between narration and declarations.

◮ in OMDoc: statements are logical paragraphs with markup for special relations ◮ in MMT: statements declarations of the form c[: τ][= δ] (via Curry-Howard Iso) ◮ Example 0.4 (Parallel Markup for a Definition)

Definition 08.15 The exponential function e· : R → R is the solution of f ′ = f with f (0) = 1. exp : R → R = τf .f ′ = f ∧ f (0) = 1

◮ Implementation: Use existing OMDoc/MMT markup as anchors, OMDoc

metadata markup framework [LK09] for crossreferencing.

Kohlhase: Flexiformality in Proofs 15 Dagstuhl

Flexiformal Proofs by Opaqueness and Parallel Markup

◮ Theorem 0.5 There are infinitely many primes

Kohlhase: Flexiformality in Proofs 16 Dagstuhl

Flexiformal Proofs by Opaqueness and Parallel Markup

◮ Theorem 0.6 There are infinitely many primes ◮ Proof Sketch:

Euclid proved this ca. 300 BC, so we leave it as an exercise

Kohlhase: Flexiformality in Proofs 16 Dagstuhl

Flexiformal Proofs by Opaqueness and Parallel Markup

◮ Theorem 0.7 There are infinitely many primes ◮ Proof Sketch:

Euclid proved this ca. 300 BC, so we leave it as an exercise

◮ Proof Sketch:

Suppose p1, p2, . . . , pk are all the primes. Then, let P = Πk

i=1pi + 1 and p a prime dividing P. But every pi divides P − 1 so p

cannot be any of them. Therefore p is a new prime. Contradiction, so there are infinitely many primes.

Kohlhase: Flexiformality in Proofs 16 Dagstuhl

Flexiformal Proofs by Opaqueness and Parallel Markup

◮ Theorem 0.8 There are infinitely many primes ◮ Proof Sketch:

Euclid proved this ca. 300 BC, so we leave it as an exercise

◮ Proof Sketch:

Suppose p1, p2, . . . , pk are all the primes. Then, let P = Πk

i=1pi + 1 and p a prime dividing P. But every pi divides P − 1 so p

cannot be any of them. Therefore p is a new prime. Contradiction, so there are infinitely many primes.

◮ Proof:

P.1 Suppose p1, p2, . . . , pk are all the primes. P.2 Then, let P = Πk

i=1pi + 1 and p a prime dividing P.

P.3 But every pi divides P − 1 so p cannot be any of them. P.4 Therefore p is a new prime. P.5 Contradiction, so there are infinitely many primes.

Kohlhase: Flexiformality in Proofs 16 Dagstuhl

Flexiformal Proofs by Opaqueness and Parallel Markup

◮ Theorem 0.9 There are infinitely many primes ◮ Proof Sketch:

Euclid proved this ca. 300 BC, so we leave it as an exercise

◮ Proof Sketch:

Suppose p1, p2, . . . , pk are all the primes. Then, let P = Πk

i=1pi + 1 and p a prime dividing P. But every pi divides P − 1 so p

cannot be any of them. Therefore p is a new prime. Contradiction, so there are infinitely many primes.

◮ Proof:

P.1 Suppose p1, p2, . . . , pk are all the primes. P.2 Then, let P = Πk

i=1pi + 1 and p a prime dividing P.

P.3 But every pi divides P − 1 so p cannot be any of them. P.4 Therefore p is a new prime. P.5 Contradiction, so there are infinitely many primes. Flexiformal Proof term: ⊥E(λX :⊢ (p1, p2, . . . , pk are . . . ). . . . ⊥)) (formal justifications, opaque types)

Kohlhase: Flexiformality in Proofs 16 Dagstuhl

Flexiformal Proofs by Opaqueness and Parallel Markup

◮ Theorem 0.10 There are infinitely many primes ◮ Proof Sketch:

Euclid proved this ca. 300 BC, so we leave it as an exercise

◮ Proof Sketch:

Suppose p1, p2, . . . , pk are all the primes. Then, let P = Πk

i=1pi + 1 and p a prime dividing P. But every pi divides P − 1 so p

cannot be any of them. Therefore p is a new prime. Contradiction, so there are infinitely many primes.

◮ Proof:

P.1 Suppose p1, p2, . . . , pk are all the primes. P.2 Then, let P = Πk

i=1pi + 1 and p a prime dividing P.

P.3 But every pi divides P − 1 so p cannot be any of them. P.4 Therefore p is a new prime. P.5 Contradiction, so there are infinitely many primes. Flexiformal Proof term: ⊥E(λX :⊢ (p1, p2, . . . , pk are . . . ). . . . ⊥)) (formal justifications, opaque types)

◮ ◮ Fully formal proof term in LF: ⊥E(λX :⊢ (. . .). . . . ⊥))

Kohlhase: Flexiformality in Proofs 16 Dagstuhl

Flexiformal Proofs by Opaqueness and Parallel Markup

◮ Theorem 0.11 There are infinitely many primes ◮ Proof Sketch:

Euclid proved this ca. 300 BC, so we leave it as an exercise

◮ Proof Sketch:

Suppose p1, p2, . . . , pk are all the primes. Then, let P = Πk

i=1pi + 1 and p a prime dividing P. But every pi divides P − 1 so p

cannot be any of them. Therefore p is a new prime. Contradiction, so there are infinitely many primes.

◮ Proof:

P.1 Suppose p1, p2, . . . , pk are all the primes. P.2 Then, let P = Πk

i=1pi + 1 and p a prime dividing P.

P.3 But every pi divides P − 1 so p cannot be any of them. P.4 Therefore p is a new prime. P.5 Contradiction, so there are infinitely many primes. Flexiformal Proof term: ⊥E(λX :⊢ (p1, p2, . . . , pk are . . . ). . . . ⊥)) (formal justifications, opaque types)

◮ ◮ Fully formal proof term in LF: ⊥E(λX :⊢ (. . .). . . . ⊥)) ◮ Cf. Claudio et al’s paper on mapping OMDoc proofs into ¯

λµ˜ µ-calculus [ASC06]

Kohlhase: Flexiformality in Proofs 16 Dagstuhl

But it can become even more opaque

◮ In my world (of theory graphs) flexiformality can appear even earlier ◮ In flexiformal views even the proof obligations can be unspecified ◮ Example 0.12 Consider the following two theories

Theory: Normed_VS If V is a vector space over F, then | · | : F → F is called a norm, iff for all a ∈ F and u, v ∈ V we have (A) |av| = |a| |v|, (T) |u + v| ≤ |u| + |v|, and (S) v = 0 if |v| = 0. Theory: Metric_Space Let M be a set, then we call a function d : M2 → R a metric on M, iff (I) d(x, y) = 0 iff x = y, (S) d(x, y) = d(y, x), and (T) d(x, z) ≤ d(x, y)+d(y, z).

◮ Example 0.13 (Empty View)

View: Metric_Space → Normed_VS this is well-known.

Kohlhase: Flexiformality in Proofs 17 Dagstuhl

But it can become even more opaque

◮ In my world (of theory graphs) flexiformality can appear even earlier ◮ In flexiformal views even the proof obligations can be unspecified ◮ Example 0.17 Consider the following two theories

Theory: Normed_VS If V is a vector space over F, then | · | : F → F is called a norm, iff for all a ∈ F and u, v ∈ V we have (A) |av| = |a| |v|, (T) |u + v| ≤ |u| + |v|, and (S) v = 0 if |v| = 0. Theory: Metric_Space Let M be a set, then we call a function d : M2 → R a metric on M, iff (I) d(x, y) = 0 iff x = y, (S) d(x, y) = d(y, x), and (T) d(x, z) ≤ d(x, y)+d(y, z).

Example 0.19 (Opaque View with partial symbol mapping)

View: Metric_Space → Normed_VS M → V, d(x, y) → |x − y|

Kohlhase: Flexiformality in Proofs 17 Dagstuhl

But it can become even more opaque

◮ In my world (of theory graphs) flexiformality can appear even earlier ◮ In flexiformal views even the proof obligations can be unspecified ◮ Example 0.22 Consider the following two theories

Theory: Normed_VS If V is a vector space over F, then | · | : F → F is called a norm, iff for all a ∈ F and u, v ∈ V we have (A) |av| = |a| |v|, (T) |u + v| ≤ |u| + |v|, and (S) v = 0 if |v| = 0. Theory: Metric_Space Let M be a set, then we call a function d : M2 → R a metric on M, iff (I) d(x, y) = 0 iff x = y, (S) d(x, y) = d(y, x), and (T) d(x, z) ≤ d(x, y)+d(y, z).

Example 0.25 (Opaque View with proof obligations)

View: Metric_Space → Normed_VS M → V, d(x, y) → |x − y|, I → A S → S , T → T

Kohlhase: Flexiformality in Proofs 17 Dagstuhl

But it can become even more opaque

◮ In my world (of theory graphs) flexiformality can appear even earlier ◮ In flexiformal views even the proof obligations can be unspecified ◮ Example 0.27 Consider the following two theories

Theory: Normed_VS If V is a vector space over F, then | · | : F → F is called a norm, iff for all a ∈ F and u, v ∈ V we have (A) |av| = |a| |v|, (T) |u + v| ≤ |u| + |v|, and (S) v = 0 if |v| = 0. Theory: Metric_Space Let M be a set, then we call a function d : M2 → R a metric on M, iff (I) d(x, y) = 0 iff x = y, (S) d(x, y) = d(y, x), and (T) d(x, z) ≤ d(x, y)+d(y, z).

Example 0.31 (Formal View) with full proof terms.

Kohlhase: Flexiformality in Proofs 17 Dagstuhl

Taking Informality Seriously in Theory Graphs

◮ Some of the contents is opaque to formal/syntactic methods Nat N, 0, 1, 2, . . . uNat N1, 0, s P1, . . . , P5 biNat N2, 0, 1, s0, s1 P1′, . . . Comp C, i arithNat +, ∗ : N∗ → N arithComp +, ∗ : C∗ → C arithPoly +, ∗ : P[C]∗ → P[C] arithNatrec ∀x : N∗.x + 0 = x ∀x : N∗, y : N.x + s(y) = s(x + y) arith0 +, ∗ alg0 e+, e∗ arith1 ∀x, y.x + y = y + x polyC P[C] ◮ Problem: But what is the meaning of the ◮ Idea: make it so that arrows are views.

(adopt the meaning)

Kohlhase: Flexiformality in Proofs 18 Dagstuhl

Conclusions & Future Work

◮ Problem (for FM): Mathematical Documents are predominantly informaly given

Formalization is expensive (need to deal with informal knowledge)

◮ Solution (OMDoc): mark up formula/statement/theory structure

Kohlhase: Flexiformality in Proofs 19 Dagstuhl

Conclusions & Future Work

◮ Problem (for FM): Mathematical Documents are predominantly informaly given

Formalization is expensive (need to deal with informal knowledge)

◮ Solution (OMDoc): mark up formula/statement/theory structure ◮ Problem: we want to retain as much as possible from Theory Graphs (MMT) ◮ Solution: Add opaque content to MMT at all levels

Kohlhase: Flexiformality in Proofs 19 Dagstuhl

Conclusions & Future Work

◮ Problem (for FM): Mathematical Documents are predominantly informaly given

Formalization is expensive (need to deal with informal knowledge)

◮ Solution (OMDoc): mark up formula/statement/theory structure ◮ Problem: we want to retain as much as possible from Theory Graphs (MMT) ◮ Solution: Add opaque content to MMT at all levels ◮ Problem: how to give meaning to opaque content? ◮ Solution 1: Allow in-place formalizations by parallel markup ◮ Solution 2: Relate to existing formalizations via postuated views/adoptions

Kohlhase: Flexiformality in Proofs 19 Dagstuhl

Conclusions & Future Work

◮ Problem (for FM): Mathematical Documents are predominantly informaly given

Formalization is expensive (need to deal with informal knowledge)

◮ Solution (OMDoc): mark up formula/statement/theory structure ◮ Problem: we want to retain as much as possible from Theory Graphs (MMT) ◮ Solution: Add opaque content to MMT at all levels ◮ Problem: how to give meaning to opaque content? ◮ Solution 1: Allow in-place formalizations by parallel markup ◮ Solution 2: Relate to existing formalizations via postuated views/adoptions ◮ Current Work: towards OMDoc2 ˆ

= narrative OMDoc + MMT

◮ do the final language design based on these ideas. ◮ case studies, case studies, case studies. Kohlhase: Flexiformality in Proofs 19 Dagstuhl

Conclusions & Future Work

◮ Problem (for FM): Mathematical Documents are predominantly informaly given

Formalization is expensive (need to deal with informal knowledge)

◮ Solution (OMDoc): mark up formula/statement/theory structure ◮ Problem: we want to retain as much as possible from Theory Graphs (MMT) ◮ Solution: Add opaque content to MMT at all levels ◮ Problem: how to give meaning to opaque content? ◮ Solution 1: Allow in-place formalizations by parallel markup ◮ Solution 2: Relate to existing formalizations via postuated views/adoptions ◮ Current Work: towards OMDoc2 ˆ

= narrative OMDoc + MMT

◮ do the final language design based on these ideas. ◮ case studies, case studies, case studies.

◮ Future Work: e.g Proof Mining from arXiv

◮ arXiv: 1,200,000 papers, Deyan Ginev found 1,392,663 instances of div_ltx_proof

(aka proofs)

Kohlhase: Flexiformality in Proofs 19 Dagstuhl

Serge Autexier and Claudio Sacerdoti Coen. A formal correspondence between omdoc with alternative proofs and the λµ˜ µ-calculus. In Jon Borwein and William M. Farmer, editors, Mathematical Knowledge Management (MKM), number 4108 in LNAI, pages 67–81. Springer Verlag, 2006. Henk Barendregt and Arjeh M. Cohen. Electronic communication of mathematics and the interaction of computer algebra systems and proof assistants. Journal of Symbolic Computation, 32:3–22, 2001. Paul-Olivier Dehaye, Mihnea Iancu, Michael Kohlhase, Alexander Konovalov, Samuel Lelièvre, Dennis Müller, Markus Pfeiffer, Florian Rabe, Nicolas M. Thiéry, and Tom Wiesing. Interoperability in the OpenDreamKit project: The math-in-the-middle approach. In Michael Kohlhase, Moa Johansson, Bruce Miller, Leonardo de Moura, and Frank Tompa, editors, Intelligent Computer Mathematics 2016, number 9791 in LNCS. Springer, 2016.

Kohlhase: Flexiformality in Proofs 19 Dagstuhl

Ingrid Daubechies, Clifford A. Lynch, Kathleen M. Carley, Timothy W. Cole, Judith L. Klavans, Yann LeCun, Michael Lesk, Peter Olver, Jim Pitman, and Zhihong Xia. Developing a 21st Century Global Library for Mathematics Research. THE NATIONAL ACADEMIES PRESS, 2014. Michael Kohlhase. OMDoc – An open markup format for mathematical documents [Version 1.2]. Number 4180 in LNAI. Springer Verlag, August 2006. Michael Kohlhase. The flexiformalist manifesto. In Andrei Voronkov, Viorel Negru, Tetsuo Ida, Tudor Jebelean, Dana Petcu, Stephen M. Watt, and Daniela Zaharie, editors, 14th International Workshop

• n Symbolic and Numeric Algorithms for Scientific Computing (SYNASC

2012), pages 30–36, Timisoara, Romania, 2013. IEEE Press. Michael Kohlhase and Florian Rabe. QED reloaded: Towards a pluralistic formal library of mathematical knowledge. Journal of Formalized Reasoning, 9(1):201–234, 2016. Christoph Lange and Michael Kohlhase.

Kohlhase: Flexiformality in Proofs 19 Dagstuhl

A mathematical approach to ontology authoring and documentation. In Jacques Carette, Lucas Dixon, Claudio Sacerdoti Coen, and Stephen M. Watt, editors, MKM/Calculemus Proceedings, number 5625 in LNAI, pages 389–404. Springer Verlag, July 2009. MMT – language and system for the uniform representation of knowledge. project web site at https://uniformal.github.io/. The OMDoc project. http://omdoc.org. Florian Rabe and Michael Kohlhase. A scalable module system. Information & Computation, 0(230):1–54, 2013. Freek Wiedijk. The “de Bruijn factor”. web page at http://www.cs.ru.nl/~freek/factor/, 2012.

Kohlhase: Flexiformality in Proofs 19 Dagstuhl