Lawrence Paulson
@lawrpaulson.bsky.social
Computer scientist with a background in mathematics and logic. Academic researching formal verification technologies and applications. Also in the cesspit
Steiners Line Theorem
A Ramos et al
For a nondegenerate triangle and a point on its circumcircle, the reflections of that point in the three sidelines are collinear, and the resulting line passes through the orthocenter. The development imports the AFP's Wallace–Simson Line entry.
A Ramos et al
For a nondegenerate triangle and a point on its circumcircle, the reflections of that point in the three sidelines are collinear, and the resulting line passes through the orthocenter. The development imports the AFP's Wallace–Simson Line entry.
Steiner's Line Theorem in Isabelle/HOL
Steiner's Line Theorem in Isabelle/HOL in the Archive of Formal Proofs
isa-afp.org
October 8, 2026 at 12:09 PM
Steiners Line Theorem
A Ramos et al
For a nondegenerate triangle and a point on its circumcircle, the reflections of that point in the three sidelines are collinear, and the resulting line passes through the orthocenter. The development imports the AFP's Wallace–Simson Line entry.
A Ramos et al
For a nondegenerate triangle and a point on its circumcircle, the reflections of that point in the three sidelines are collinear, and the resulting line passes through the orthocenter. The development imports the AFP's Wallace–Simson Line entry.
The Steiner Deltoid as the Tangent Envelope of Wallace–Simson Lines
A Ramos et al
In unit-circumcircle coordinates, the deltoid has an explicit complex parametrisation. We prove its derivative formula and an equality of sets between the appropriate Wallace–Simson line and the tangent line.
A Ramos et al
In unit-circumcircle coordinates, the deltoid has an explicit complex parametrisation. We prove its derivative formula and an equality of sets between the appropriate Wallace–Simson line and the tangent line.
The Steiner Deltoid as the Tangent Envelope of Wallace--Simson Lines in Isabelle/HOL
The Steiner Deltoid as the Tangent Envelope of Wallace--Simson Lines in Isabelle/HOL in the Archive of Formal Proofs
isa-afp.org
October 7, 2026 at 5:33 PM
The Steiner Deltoid as the Tangent Envelope of Wallace–Simson Lines
A Ramos et al
In unit-circumcircle coordinates, the deltoid has an explicit complex parametrisation. We prove its derivative formula and an equality of sets between the appropriate Wallace–Simson line and the tangent line.
A Ramos et al
In unit-circumcircle coordinates, the deltoid has an explicit complex parametrisation. We prove its derivative formula and an equality of sets between the appropriate Wallace–Simson line and the tangent line.
EA hits the cover of the Economist. Why are so many philosophers bonkers?
October 1, 2026 at 8:58 PM
EA hits the cover of the Economist. Why are so many philosophers bonkers?
General Weierstrass Equations
AF Ramos et al.
This entry develops the general Weierstrass equation over commutative rings and fields. The definitions and invariant formulas follow the standard treatment in Silverman.
isa-afp.org/entries/Gene...
AF Ramos et al.
This entry develops the general Weierstrass equation over commutative rings and fields. The definitions and invariant formulas follow the standard treatment in Silverman.
isa-afp.org/entries/Gene...
General Weierstrass Equations
General Weierstrass Equations in the Archive of Formal Proofs
isa-afp.org
October 1, 2026 at 3:27 PM
General Weierstrass Equations
AF Ramos et al.
This entry develops the general Weierstrass equation over commutative rings and fields. The definitions and invariant formulas follow the standard treatment in Silverman.
isa-afp.org/entries/Gene...
AF Ramos et al.
This entry develops the general Weierstrass equation over commutative rings and fields. The definitions and invariant formulas follow the standard treatment in Silverman.
isa-afp.org/entries/Gene...
Work Bounds for Strongly Joinable Balanced Binary Search Trees
M Haucke
The complexity analysis of set operations on strongly joinable trees by Blelloch, Ferizovic and Sun: union, intersection, and difference run in time O(m log (n/m +1)) for input sets of size n and m where m≤n.
M Haucke
The complexity analysis of set operations on strongly joinable trees by Blelloch, Ferizovic and Sun: union, intersection, and difference run in time O(m log (n/m +1)) for input sets of size n and m where m≤n.
Work Bounds for Strongly Joinable Balanced Binary Search Trees
Work Bounds for Strongly Joinable Balanced Binary Search Trees in the Archive of Formal Proofs
isa-afp.org
September 25, 2026 at 11:04 AM
Work Bounds for Strongly Joinable Balanced Binary Search Trees
M Haucke
The complexity analysis of set operations on strongly joinable trees by Blelloch, Ferizovic and Sun: union, intersection, and difference run in time O(m log (n/m +1)) for input sets of size n and m where m≤n.
M Haucke
The complexity analysis of set operations on strongly joinable trees by Blelloch, Ferizovic and Sun: union, intersection, and difference run in time O(m log (n/m +1)) for input sets of size n and m where m≤n.
Deterministic Context-Free Languages are Closed Under Complementation
K Taskin, T Nipkow
A deterministic context-free language (DCFL) is a language accepted by a deterministic pushdown automaton This entry proves that they are closed under complementation, following Hopcroft and Ullman (1979).
K Taskin, T Nipkow
A deterministic context-free language (DCFL) is a language accepted by a deterministic pushdown automaton This entry proves that they are closed under complementation, following Hopcroft and Ullman (1979).
Deterministic Context-Free Languages are Closed Under Complementation (Hopcroft and Ullman)
Deterministic Context-Free Languages are Closed Under Complementation (Hopcroft and Ullman) in the Archive of Formal Proofs
isa-afp.org
September 24, 2026 at 1:45 PM
Deterministic Context-Free Languages are Closed Under Complementation
K Taskin, T Nipkow
A deterministic context-free language (DCFL) is a language accepted by a deterministic pushdown automaton This entry proves that they are closed under complementation, following Hopcroft and Ullman (1979).
K Taskin, T Nipkow
A deterministic context-free language (DCFL) is a language accepted by a deterministic pushdown automaton This entry proves that they are closed under complementation, following Hopcroft and Ullman (1979).
The Teichmüller-Tukey Lemma
V Kraisch, L Cordeiro
This entry formalizes the Teichmüller-Tukey lemma: every nonempty family of sets of finite character contains a member that is maximal under inclusion. The development follows the direct choice-function construction of Sun and Yu.
V Kraisch, L Cordeiro
This entry formalizes the Teichmüller-Tukey lemma: every nonempty family of sets of finite character contains a member that is maximal under inclusion. The development follows the direct choice-function construction of Sun and Yu.
The Teichmüller-Tukey Lemma
The Teichmüller-Tukey Lemma in the Archive of Formal Proofs
isa-afp.org
September 23, 2026 at 10:34 AM
The Teichmüller-Tukey Lemma
V Kraisch, L Cordeiro
This entry formalizes the Teichmüller-Tukey lemma: every nonempty family of sets of finite character contains a member that is maximal under inclusion. The development follows the direct choice-function construction of Sun and Yu.
V Kraisch, L Cordeiro
This entry formalizes the Teichmüller-Tukey lemma: every nonempty family of sets of finite character contains a member that is maximal under inclusion. The development follows the direct choice-function construction of Sun and Yu.
Strong Normalization for Church-Style System F
AF Ramos et al.
Every well-typed System F term is strongly normalizing. The reduction relation contains both term-beta and type-beta steps and is closed under all term contexts. The proof uses Girard-style reducibility candidates.
AF Ramos et al.
Every well-typed System F term is strongly normalizing. The reduction relation contains both term-beta and type-beta steps and is closed under all term contexts. The proof uses Girard-style reducibility candidates.
Strong Normalization for Church-Style System F
Strong Normalization for Church-Style System F in the Archive of Formal Proofs
isa-afp.org
September 18, 2026 at 9:40 AM
Strong Normalization for Church-Style System F
AF Ramos et al.
Every well-typed System F term is strongly normalizing. The reduction relation contains both term-beta and type-beta steps and is closed under all term contexts. The proof uses Girard-style reducibility candidates.
AF Ramos et al.
Every well-typed System F term is strongly normalizing. The reduction relation contains both term-beta and type-beta steps and is closed under all term contexts. The proof uses Girard-style reducibility candidates.
Greibach's Hardest Context-Free Language
by Tobias Nipkow
A formalization of Greibach’s hardest context-free language theorem: There is a “hardest” context-free language L_0 such that every context-free language is an inverse homomorphic image of L_0.
isa-afp.org/entries/Grei...
by Tobias Nipkow
A formalization of Greibach’s hardest context-free language theorem: There is a “hardest” context-free language L_0 such that every context-free language is an inverse homomorphic image of L_0.
isa-afp.org/entries/Grei...
Greibach’s Hardest Context-Free Language
Greibach’s Hardest Context-Free Language in the Archive of Formal Proofs
isa-afp.org
September 17, 2026 at 4:59 PM
Greibach's Hardest Context-Free Language
by Tobias Nipkow
A formalization of Greibach’s hardest context-free language theorem: There is a “hardest” context-free language L_0 such that every context-free language is an inverse homomorphic image of L_0.
isa-afp.org/entries/Grei...
by Tobias Nipkow
A formalization of Greibach’s hardest context-free language theorem: There is a “hardest” context-free language L_0 such that every context-free language is an inverse homomorphic image of L_0.
isa-afp.org/entries/Grei...
The Földes–Hammer Characterization of Split Graphs
by AF Ramos et al
This entry formalizes the classical Földes–Hammer theorem characterizing finite split graphs
as exactly the graphs with no induced copy of 2 K_2, C_4, or C_5.
isa-afp.org/entries/Fold...
by AF Ramos et al
This entry formalizes the classical Földes–Hammer theorem characterizing finite split graphs
as exactly the graphs with no induced copy of 2 K_2, C_4, or C_5.
isa-afp.org/entries/Fold...
isa-afp.org
September 12, 2026 at 10:12 AM
The Földes–Hammer Characterization of Split Graphs
by AF Ramos et al
This entry formalizes the classical Földes–Hammer theorem characterizing finite split graphs
as exactly the graphs with no induced copy of 2 K_2, C_4, or C_5.
isa-afp.org/entries/Fold...
by AF Ramos et al
This entry formalizes the classical Földes–Hammer theorem characterizing finite split graphs
as exactly the graphs with no induced copy of 2 K_2, C_4, or C_5.
isa-afp.org/entries/Fold...
The Five Platonic Solids
E Finken
There are exactly five Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. A Platonic solid is a convex polyhedron whose faces are congruent regular polygons, with the same number of edges meeting at each vertex.
E Finken
There are exactly five Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. A Platonic solid is a convex polyhedron whose faces are congruent regular polygons, with the same number of edges meeting at each vertex.
The Five Platonic Solids
The Five Platonic Solids in the Archive of Formal Proofs
isa-afp.org
September 11, 2026 at 7:49 PM
The Five Platonic Solids
E Finken
There are exactly five Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. A Platonic solid is a convex polyhedron whose faces are congruent regular polygons, with the same number of edges meeting at each vertex.
E Finken
There are exactly five Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. A Platonic solid is a convex polyhedron whose faces are congruent regular polygons, with the same number of edges meeting at each vertex.
A Z Salamon and M Wehar have given us a large chunk of multitape Turing machine theory:
isa-afp.org/entries/Mult...
isa-afp.org/entries/Mult...
isa-afp.org/entries/Mult...
isa-afp.org/entries/Mult...
If you ever wondered how some TM construction in Hopcroft and Ullman really works, look no further
isa-afp.org/entries/Mult...
isa-afp.org/entries/Mult...
isa-afp.org/entries/Mult...
isa-afp.org/entries/Mult...
If you ever wondered how some TM construction in Hopcroft and Ullman really works, look no further
Multitape Turing Machine Substrate
Multitape Turing Machine Substrate in the Archive of Formal Proofs
isa-afp.org
August 26, 2026 at 8:47 AM
A Z Salamon and M Wehar have given us a large chunk of multitape Turing machine theory:
isa-afp.org/entries/Mult...
isa-afp.org/entries/Mult...
isa-afp.org/entries/Mult...
isa-afp.org/entries/Mult...
If you ever wondered how some TM construction in Hopcroft and Ullman really works, look no further
isa-afp.org/entries/Mult...
isa-afp.org/entries/Mult...
isa-afp.org/entries/Mult...
isa-afp.org/entries/Mult...
If you ever wondered how some TM construction in Hopcroft and Ullman really works, look no further
Miquel's Theorem
A Ramos
Let ABC be a triangle in the Euclidean plane and let P, Q, R be points on the side lines BC, CA, AB respectively. Then the circumcircles of the three triangles AQR, BRP, CPQ pass through a common point, the Miquel point of the configuration.
isa-afp.org/entries/Miqu...
A Ramos
Let ABC be a triangle in the Euclidean plane and let P, Q, R be points on the side lines BC, CA, AB respectively. Then the circumcircles of the three triangles AQR, BRP, CPQ pass through a common point, the Miquel point of the configuration.
isa-afp.org/entries/Miqu...
Miquel's Theorem in Isabelle/HOL
Miquel's Theorem in Isabelle/HOL in the Archive of Formal Proofs
isa-afp.org
August 24, 2026 at 8:19 PM
Miquel's Theorem
A Ramos
Let ABC be a triangle in the Euclidean plane and let P, Q, R be points on the side lines BC, CA, AB respectively. Then the circumcircles of the three triangles AQR, BRP, CPQ pass through a common point, the Miquel point of the configuration.
isa-afp.org/entries/Miqu...
A Ramos
Let ABC be a triangle in the Euclidean plane and let P, Q, R be points on the side lines BC, CA, AB respectively. Then the circumcircles of the three triangles AQR, BRP, CPQ pass through a common point, the Miquel point of the configuration.
isa-afp.org/entries/Miqu...
Laurent Series Expansions on an Annulus
M Eberl
A complex-valued function holomorphic on an open annulus has a (generalised) Laurent series expansion, which is valid on the entire annulus. An important special case is r=0, i.e. the local behaviour of a function around an isolated singularity.
M Eberl
A complex-valued function holomorphic on an open annulus has a (generalised) Laurent series expansion, which is valid on the entire annulus. An important special case is r=0, i.e. the local behaviour of a function around an isolated singularity.
Laurent Series Expansions on an Annulus
Laurent Series Expansions on an Annulus in the Archive of Formal Proofs
isa-afp.org
August 23, 2026 at 10:36 AM
Laurent Series Expansions on an Annulus
M Eberl
A complex-valued function holomorphic on an open annulus has a (generalised) Laurent series expansion, which is valid on the entire annulus. An important special case is r=0, i.e. the local behaviour of a function around an isolated singularity.
M Eberl
A complex-valued function holomorphic on an open annulus has a (generalised) Laurent series expansion, which is valid on the entire annulus. An important special case is r=0, i.e. the local behaviour of a function around an isolated singularity.
How I came to write THAT paper with Leslie Lamport
lawrencecpaulson.github.io
August 21, 2026 at 4:44 PM
First-Order Methods for Smooth Convex Optimization
Feier Lyu
General-purpose interfaces for gradients, first-order convexity certificates, smooth quadratic upper bounds, descent and telescoping arguments, projection geometry, projected-gradient
mappings, residual certificates, etc.
Feier Lyu
General-purpose interfaces for gradients, first-order convexity certificates, smooth quadratic upper bounds, descent and telescoping arguments, projection geometry, projected-gradient
mappings, residual certificates, etc.
August 13, 2026 at 2:36 PM
First-Order Methods for Smooth Convex Optimization
Feier Lyu
General-purpose interfaces for gradients, first-order convexity certificates, smooth quadratic upper bounds, descent and telescoping arguments, projection geometry, projected-gradient
mappings, residual certificates, etc.
Feier Lyu
General-purpose interfaces for gradients, first-order convexity certificates, smooth quadratic upper bounds, descent and telescoping arguments, projection geometry, projected-gradient
mappings, residual certificates, etc.
Reposted by Lawrence Paulson
Counterexample to Cost-Preserving Single-Source Unsplittable Flow
A Ramos et al
Dinitz, Garg, and Goemans proved that a feasible fractional single-source flow can be rounded to an unsplittable flow with additive congestion bounded. Goemans conjectured that the rounding can also preserve cost.
A Ramos et al
Dinitz, Garg, and Goemans proved that a feasible fractional single-source flow can be rounded to an unsplittable flow with additive congestion bounded. Goemans conjectured that the rounding can also preserve cost.
A Formal Counterexample to the Cost-Preserving Single-Source Unsplittable Flow Conjecture
A Formal Counterexample to the Cost-Preserving Single-Source Unsplittable Flow Conjecture in the Archive of Formal Proofs
isa-afp.org
August 7, 2026 at 2:31 PM
Counterexample to Cost-Preserving Single-Source Unsplittable Flow
A Ramos et al
Dinitz, Garg, and Goemans proved that a feasible fractional single-source flow can be rounded to an unsplittable flow with additive congestion bounded. Goemans conjectured that the rounding can also preserve cost.
A Ramos et al
Dinitz, Garg, and Goemans proved that a feasible fractional single-source flow can be rounded to an unsplittable flow with additive congestion bounded. Goemans conjectured that the rounding can also preserve cost.
Completeness of the Q0 Higher-Order Logic
AH From et al
Our work builds on Díaz's formalization of Q0's syntax, semantics, soundness and consistency. We prove completeness using the framework for abstract consistency properties by From and Schlichtkrull to get a model existence theorem for Q0.
AH From et al
Our work builds on Díaz's formalization of Q0's syntax, semantics, soundness and consistency. We prove completeness using the framework for abstract consistency properties by From and Schlichtkrull to get a model existence theorem for Q0.
Completeness of the Q0 Higher-Order Logic
Completeness of the Q0 Higher-Order Logic in the Archive of Formal Proofs
isa-afp.org
August 6, 2026 at 4:06 PM
Completeness of the Q0 Higher-Order Logic
AH From et al
Our work builds on Díaz's formalization of Q0's syntax, semantics, soundness and consistency. We prove completeness using the framework for abstract consistency properties by From and Schlichtkrull to get a model existence theorem for Q0.
AH From et al
Our work builds on Díaz's formalization of Q0's syntax, semantics, soundness and consistency. We prove completeness using the framework for abstract consistency properties by From and Schlichtkrull to get a model existence theorem for Q0.
I've uploaded a new video on 40 Years of Isabelle. Apologies for the lousy thumbnail.
youtu.be/nEf-WVtpEek
youtu.be/nEf-WVtpEek
Isabelle 40 Years
YouTube video by Lawrence Paulson
youtu.be
August 6, 2026 at 3:55 PM
I've uploaded a new video on 40 Years of Isabelle. Apologies for the lousy thumbnail.
youtu.be/nEf-WVtpEek
youtu.be/nEf-WVtpEek
The Wallace–Simson Line Theorem
AF Ramos et al
Let ABC be a triangle and M a point on its circumcircle. Dropping the perpendiculars from M to the three side lines gives feet P, Q, R; then P,Q,R are collinear. Conversely if the three feet are collinear then M lies on the circumcircle of ABC.
AF Ramos et al
Let ABC be a triangle and M a point on its circumcircle. Dropping the perpendiculars from M to the three side lines gives feet P, Q, R; then P,Q,R are collinear. Conversely if the three feet are collinear then M lies on the circumcircle of ABC.
The Wallace--Simson Line Theorem in Isabelle/HOL
The Wallace--Simson Line Theorem in Isabelle/HOL in the Archive of Formal Proofs
isa-afp.org
August 5, 2026 at 2:39 PM
The Wallace–Simson Line Theorem
AF Ramos et al
Let ABC be a triangle and M a point on its circumcircle. Dropping the perpendiculars from M to the three side lines gives feet P, Q, R; then P,Q,R are collinear. Conversely if the three feet are collinear then M lies on the circumcircle of ABC.
AF Ramos et al
Let ABC be a triangle and M a point on its circumcircle. Dropping the perpendiculars from M to the three side lines gives feet P, Q, R; then P,Q,R are collinear. Conversely if the three feet are collinear then M lies on the circumcircle of ABC.
New on my blog: "Why is it all in the kernel?"
lawrencecpaulson.github.io/2026/07/30/C...
lawrencecpaulson.github.io/2026/07/30/C...
Why is it all in the kernel?
lawrencecpaulson.github.io
July 30, 2026 at 8:28 PM
New on my blog: "Why is it all in the kernel?"
lawrencecpaulson.github.io/2026/07/30/C...
lawrencecpaulson.github.io/2026/07/30/C...
Counterexample to the Jacobian Conjecture
AF Ramos et al
The conjecture asks whether a polynomial self-map of affine space has a polynomial inverse whenever its Jacobian determinant is a nonzero constant. This entry gives an Isabelle/HOL verification of the 3-d map announced by Levent Alpöge.
AF Ramos et al
The conjecture asks whether a polynomial self-map of affine space has a polynomial inverse whenever its Jacobian determinant is a nonzero constant. This entry gives an Isabelle/HOL verification of the 3-d map announced by Levent Alpöge.
Formal Verification of an Explicit Counterexample to the Jacobian Conjecture
Formal Verification of an Explicit Counterexample to the Jacobian Conjecture in the Archive of Formal Proofs
isa-afp.org
July 21, 2026 at 8:04 PM
Counterexample to the Jacobian Conjecture
AF Ramos et al
The conjecture asks whether a polynomial self-map of affine space has a polynomial inverse whenever its Jacobian determinant is a nonzero constant. This entry gives an Isabelle/HOL verification of the 3-d map announced by Levent Alpöge.
AF Ramos et al
The conjecture asks whether a polynomial self-map of affine space has a polynomial inverse whenever its Jacobian determinant is a nonzero constant. This entry gives an Isabelle/HOL verification of the 3-d map announced by Levent Alpöge.
