First-Order Deduction for Large Knowledge Bases Stephan Schulz - - PowerPoint PPT Presentation

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First-Order Deduction for Large Knowledge Bases Stephan Schulz - - PowerPoint PPT Presentation

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First-Order Deduction for Large Knowledge Bases Stephan Schulz schulz@eprover.org E Stephan Schulz Interest in Large Theories 2007 ESARLT at CADE-21 in Bremen Ontologies (SUMO) Common Sense Reasoning (CYC) Mizar 2008 First

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First-Order Deduction for Large Knowledge Bases

Stephan Schulz schulz@eprover.org

E

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Stephan Schulz

Interest in Large Theories

2007 ESARLT at CADE-21 in Bremen

◮ Ontologies (SUMO) ◮ Common Sense Reasoning (CYC) ◮ Mizar

2008 First CASC LTB at IJCAR in Sydney

◮ MaLARea, SInE, Vampire LTB

2009 CASC LTB at CADE-22 in Montreal 2010 CASC LTB in IJCAR in Edinburgh

◮ Rules change every year!

2

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SLIDE 3

Stephan Schulz

Interest in Large Theories

2007 ESARLT at CADE-21 in Bremen

◮ Ontologies (SUMO) ◮ Common Sense Reasoning (CYC) ◮ Mizar

2008 First CASC LTB at IJCAR in Sydney

◮ MaLARea, SInE, Vampire LTB

2009 CASC LTB at CADE-22 in Montreal 2010 CASC LTB in IJCAR in Edinburgh

◮ Rules change every year!

2

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SLIDE 4

Stephan Schulz

Interest in Large Theories

2007 ESARLT at CADE-21 in Bremen

◮ Ontologies (SUMO) ◮ Common Sense Reasoning (CYC) ◮ Mizar

2008 First CASC LTB at IJCAR in Sydney

◮ MaLARea, SInE, Vampire LTB

2009 CASC LTB at CADE-22 in Montreal 2010 CASC LTB in IJCAR in Edinburgh

◮ Rules change every year!

2

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SLIDE 5

Stephan Schulz

Interest in Large Theories

2007 ESARLT at CADE-21 in Bremen

◮ Ontologies (SUMO) ◮ Common Sense Reasoning (CYC) ◮ Mizar

2008 First CASC LTB at IJCAR in Sydney

◮ MaLARea, SInE, Vampire LTB

2009 CASC LTB at CADE-22 in Montreal 2010 CASC LTB in IJCAR in Edinburgh

◮ Rules change every year!

2

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Stephan Schulz

Saturation-Based Theorem Proving

Leading approach for first-order reasoning

◮ Proof by refutation ◮ Conjecture and axioms converted to flat CNF ◮ Saturation with destructive simplification ◮ Goal: Empty clause ◮ Powerful calculi ◮ Highly advanced data structures ◮ Decent search heuristics Clausification Saturation

Axioms Proof (?) Conjecture

CNF CNF

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Stephan Schulz

A Classical Example: Reasoning in Rings

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Stephan Schulz

%----Right identity and inverse cnf(right_identity,axiom, ( add(X,additive_identity) = X )). cnf(right_additive_inverse,axiom, ( add(X,additive_inverse(X)) = additive_identity )). %----Distributive property of product over sum cnf(distribute1,axiom, ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )). cnf(distribute2,axiom, ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )). %----Associativity of addition cnf(associative_addition,axiom, ( add(add(X,Y),Z) = add(X,add(Y,Z)) )). %----Commutativity of addition cnf(commutative_addition,axiom, ( add(X,Y) = add(Y,X) )). %----Associativity of product cnf(associative_multiplication,axiom, ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )). cnf(x_cubed_is_x,hypothesis, ( multiply(X,multiply(X,X)) = X )). cnf(prove_commutativity,negated_conjecture, ( multiply(a,b) != multiply(b,a) )).

A ring with x3=x is commutative

  • 9 formulas
  • 3 KB of text (with

comments)

Proof search

  • ~200000 steps
  • ~50 MB of text

RNG009-5: Peterson/Stickel, 1981 5

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Stephan Schulz

%----Right identity and inverse cnf(right_identity,axiom, ( add(X,additive_identity) = X )). cnf(right_additive_inverse,axiom, ( add(X,additive_inverse(X)) = additive_identity )). %----Distributive property of product over sum cnf(distribute1,axiom, ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )). cnf(distribute2,axiom, ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )). %----Associativity of addition cnf(associative_addition,axiom, ( add(add(X,Y),Z) = add(X,add(Y,Z)) )). %----Commutativity of addition cnf(commutative_addition,axiom, ( add(X,Y) = add(Y,X) )). %----Associativity of product cnf(associative_multiplication,axiom, ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )). cnf(x_cubed_is_x,hypothesis, ( multiply(X,multiply(X,X)) = X )). cnf(prove_commutativity,negated_conjecture, ( multiply(a,b) != multiply(b,a) )).

A ring with x3=x is commutative

  • 9 formulas
  • 3 KB of text (with

comments)

Proof search

  • ~200000 steps
  • ~50 MB of text

RNG009-5: Peterson/Stickel, 1981 5

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SLIDE 10

Stephan Schulz

%----Right identity and inverse cnf(right_identity,axiom, ( add(X,additive_identity) = X )). cnf(right_additive_inverse,axiom, ( add(X,additive_inverse(X)) = additive_identity )). %----Distributive property of product over sum cnf(distribute1,axiom, ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )). cnf(distribute2,axiom, ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )). %----Associativity of addition cnf(associative_addition,axiom, ( add(add(X,Y),Z) = add(X,add(Y,Z)) )). %----Commutativity of addition cnf(commutative_addition,axiom, ( add(X,Y) = add(Y,X) )). %----Associativity of product cnf(associative_multiplication,axiom, ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )). cnf(x_cubed_is_x,hypothesis, ( multiply(X,multiply(X,X)) = X )). cnf(prove_commutativity,negated_conjecture, ( multiply(a,b) != multiply(b,a) )).

A ring with x3=x is commutative

  • 9 formulas
  • 3 KB of text (with

comments)

Proof search

  • ~200000 steps
  • ~50 MB of text

RNG009-5: Peterson/Stickel, 1981 5

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Stephan Schulz

%----Right identity and inverse cnf(right_identity,axiom, ( add(X,additive_identity) = X )). cnf(right_additive_inverse,axiom, ( add(X,additive_inverse(X)) = additive_identity )). %----Distributive property of product over sum cnf(distribute1,axiom, ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )). cnf(distribute2,axiom, ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )). %----Associativity of addition cnf(associative_addition,axiom, ( add(add(X,Y),Z) = add(X,add(Y,Z)) )). %----Commutativity of addition cnf(commutative_addition,axiom, ( add(X,Y) = add(Y,X) )). %----Associativity of product cnf(associative_multiplication,axiom, ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )). cnf(x_cubed_is_x,hypothesis, ( multiply(X,multiply(X,X)) = X )). cnf(prove_commutativity,negated_conjecture, ( multiply(a,b) != multiply(b,a) )).

A ring with x3=x is commutative

  • 9 formulas
  • 3 KB of text (with

comments)

Proof search

  • ~200000 steps
  • ~50 MB of text

RNG009-5: Peterson/Stickel, 1981 5

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Stephan Schulz

The Matter of Scaling

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Stephan Schulz

%----Right identity and inverse cnf(right_identity,axiom, ( add(X,additive_identity) = X )). cnf(right_additive_inverse,axiom, ( add(X,additive_inverse(X)) = additive_identity )). %----Distributive property of product over sum cnf(distribute1,axiom, ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )). cnf(distribute2,axiom, ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )). %----Associativity of addition cnf(associative_addition,axiom, ( add(add(X,Y),Z) = add(X,add(Y,Z)) )). %----Commutativity of addition cnf(commutative_addition,axiom, ( add(X,Y) = add(Y,X) )). %----Associativity of product cnf(associative_multiplication,axiom, ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )). cnf(x_cubed_is_x,hypothesis, ( multiply(X,multiply(X,X)) = X )). cnf(prove_commutativity,negated_conjecture, ( multiply(a,b) != multiply(b,a) )).

RNG009-5: Peterson/Stickel, 1981 7

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Stephan Schulz

%----Right identity and inverse cnf(right_identity,axiom, ( add(X,additive_identity) = X )). cnf(right_additive_inverse,axiom, ( add(X,additive_inverse(X)) = additive_identity )). %----Distributive property of product over sum cnf(distribute1,axiom, ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )). cnf(distribute2,axiom, ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )). %----Associativity of addition cnf(associative_addition,axiom, ( add(add(X,Y),Z) = add(X,add(Y,Z)) )). %----Commutativity of addition cnf(commutative_addition,axiom, ( add(X,Y) = add(Y,X) )). %----Associativity of product cnf(associative_multiplication,axiom, ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )). cnf(x_cubed_is_x,hypothesis, ( multiply(X,multiply(X,X)) = X )). cnf(prove_commutativity,negated_conjecture, ( multiply(a,b) != multiply(b,a) )).

RNG009-5: Peterson/Stickel, 1981 7

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Stephan Schulz

%----Right identity and inverse cnf(right_identity,axiom, ( add(X,additive_identity) = X )). cnf(right_additive_inverse,axiom, ( add(X,additive_inverse(X)) = additive_identity )). %----Distributive property of product over sum cnf(distribute1,axiom, ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )). cnf(distribute2,axiom, ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )). %----Associativity of addition cnf(associative_addition,axiom, ( add(add(X,Y),Z) = add(X,add(Y,Z)) )). %----Commutativity of addition cnf(commutative_addition,axiom, ( add(X,Y) = add(Y,X) )). %----Associativity of product cnf(associative_multiplication,axiom, ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )). cnf(x_cubed_is_x,hypothesis, ( multiply(X,multiply(X,X)) = X )). cnf(prove_commutativity,negated_conjecture, ( multiply(a,b) != multiply(b,a) )).

RNG009-5: Peterson/Stickel, 1981 7

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Stephan Schulz

%----Right identity and inverse cnf(right_identity,axiom, ( add(X,additive_identity) = X )). cnf(right_additive_inverse,axiom, ( add(X,additive_inverse(X)) = additive_identity )). %----Distributive property of product over sum cnf(distribute1,axiom, ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )). cnf(distribute2,axiom, ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )). %----Associativity of addition cnf(associative_addition,axiom, ( add(add(X,Y),Z) = add(X,add(Y,Z)) )). %----Commutativity of addition cnf(commutative_addition,axiom, ( add(X,Y) = add(Y,X) )). %----Associativity of product cnf(associative_multiplication,axiom, ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )). cnf(x_cubed_is_x,hypothesis, ( multiply(X,multiply(X,X)) = X )). cnf(prove_commutativity,negated_conjecture, ( multiply(a,b) != multiply(b,a) )).

RNG009-5: Peterson/Stickel, 1981 7

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Stephan Schulz

%----Right identity and inverse cnf(right_identity,axiom, ( add(X,additive_identity) = X )). cnf(right_additive_inverse,axiom, ( add(X,additive_inverse(X)) = additive_identity )). %----Distributive property of product over sum cnf(distribute1,axiom, ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )). cnf(distribute2,axiom, ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )). %----Associativity of addition cnf(associative_addition,axiom, ( add(add(X,Y),Z) = add(X,add(Y,Z)) )). %----Commutativity of addition cnf(commutative_addition,axiom, ( add(X,Y) = add(Y,X) )). %----Associativity of product cnf(associative_multiplication,axiom, ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )). cnf(x_cubed_is_x,hypothesis, ( multiply(X,multiply(X,X)) = X )). cnf(prove_commutativity,negated_conjecture, ( multiply(a,b) != multiply(b,a) )).

RNG009-5: Peterson/Stickel, 1981 7

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Stephan Schulz

%----Right identity and inverse cnf(right_identity,axiom, ( add(X,additive_identity) = X )). cnf(right_additive_inverse,axiom, ( add(X,additive_inverse(X)) = additive_identity )). %----Distributive property of product over sum cnf(distribute1,axiom, ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )). cnf(distribute2,axiom, ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )). %----Associativity of addition cnf(associative_addition,axiom, ( add(add(X,Y),Z) = add(X,add(Y,Z)) )). %----Commutativity of addition cnf(commutative_addition,axiom, ( add(X,Y) = add(Y,X) )). %----Associativity of product cnf(associative_multiplication,axiom, ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )). cnf(x_cubed_is_x,hypothesis, ( multiply(X,multiply(X,X)) = X )). cnf(prove_commutativity,negated_conjecture, ( multiply(a,b) != multiply(b,a) )).

RNG009-5: Peterson/Stickel, 1981 7

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Stephan Schulz

%----Right identity and inverse cnf(right_identity,axiom, ( add(X,additive_identity) = X )). cnf(right_additive_inverse,axiom, ( add(X,additive_inverse(X)) = additive_identity )). %----Distributive property of product over sum cnf(distribute1,axiom, ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )). cnf(distribute2,axiom, ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )). %----Associativity of addition cnf(associative_addition,axiom, ( add(add(X,Y),Z) = add(X,add(Y,Z)) )). %----Commutativity of addition cnf(commutative_addition,axiom, ( add(X,Y) = add(Y,X) )). %----Associativity of product cnf(associative_multiplication,axiom, ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )). cnf(x_cubed_is_x,hypothesis, ( multiply(X,multiply(X,X)) = X )). cnf(prove_commutativity,negated_conjecture, ( multiply(a,b) != multiply(b,a) )).

RNG009-5: Peterson/Stickel, 1981 7

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Stephan Schulz

%----Right identity and inverse cnf(right_identity,axiom, ( add(X,additive_identity) = X )). cnf(right_additive_inverse,axiom, ( add(X,additive_inverse(X)) = additive_identity )). %----Distributive property of product over sum cnf(distribute1,axiom, ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )). cnf(distribute2,axiom, ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )). %----Associativity of addition cnf(associative_addition,axiom, ( add(add(X,Y),Z) = add(X,add(Y,Z)) )). %----Commutativity of addition cnf(commutative_addition,axiom, ( add(X,Y) = add(Y,X) )). %----Associativity of product cnf(associative_multiplication,axiom, ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )). cnf(x_cubed_is_x,hypothesis, ( multiply(X,multiply(X,X)) = X )). cnf(prove_commutativity,negated_conjecture, ( multiply(a,b) != multiply(b,a) )).

RNG009-5: Peterson/Stickel, 1981 CSR066-6: Smith et al, 2007 7

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Stephan Schulz CSR066-6: Smith et all, 2007

OpenCYC “Common sense” reasoning

  • 3341990 formulas
  • 480 MB of text (with

few comments)

7

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Stephan Schulz CSR066-6: Smith et all, 2007

OpenCYC “Common sense” reasoning

  • 3341990 formulas
  • 480 MB of text (with

few comments)

Proof search

?

7

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Stephan Schulz

Bad Ideas for Mega-Axiom Problems

“First simplify the whole specification” “First consider all axioms” Quadratic or cubic algorithms Undirected saturation Reload the specification for each new conjecture

8

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Stephan Schulz

Multiple Queries the Hard Way

Clausification Saturation

Axioms Proof (?) Conjecture

CNF CNF Clausification Saturation

Axioms Proof (?) Conjecture

CNF CNF Clausification Saturation

Axioms Proof (?) Conjecture

CNF CNF Clausification Saturation

Axioms Proof (?) Conjecture

CNF CNF Clausification Saturation

Axioms Proof (?) Conjecture

CNF CNF Clausification Saturation

Axioms Proof (?) Conjecture

CNF CNF

Time

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Stephan Schulz

Deduction as a Service

Time Axioms

Conjecture

Deduction Server

Conjecture Conjecture Conjecture Conjecture Conjecture Conjecture Conjecture Proof (!) Proof (!) Proof (!) Proof (!) Proof (!) Proof (!) Proof (!) Proof (!)

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Stephan Schulz

Deduction Server Architecture

Clausification Saturation CNF CNF

Proof Conjecture

Axioms

Client

Axiom Selector Knowledge Base

Server

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Stephan Schulz

Deduction Server Architecture

Clausification Saturation CNF CNF Clausification Saturation CNF CNF Clausification Saturation CNF CNF

Proof Conjecture

Axioms

Client

Axiom Selector Knowledge Base

Server

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Stephan Schulz

Current State

Clausification Saturation CNF CNF Clausification Saturation CNF CNF Clausification Saturation CNF CNF

Conjecture

Axioms

Client

Axiom Selector Knowledge Base

Server

Proof

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Stephan Schulz

Axiom Selection

Conjecture Axiom Selector Knowledge Base

  • Domain axioms
  • Indexed
  • Large

Conjecture +Axioms

Input

◮ Conjecture ◮ Optionally hypotheses

Output

◮ “Relevant” axioms ◮ Conjecture

Constraints

◮ Backtrackable ◮ Fast (enough)

13

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Stephan Schulz

Relevancy Analysis

Algorithm:

◮ Start with the conjecture ◮ All symbols in the conjecture are relevant ◮ All formulas containing relevant symbols are relevant ◮ Iterate for n steps or up to fixpoint

14

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Stephan Schulz

Relevancy Analysis

Algorithm:

◮ Start with the conjecture ◮ All symbols in the conjecture are relevant ◮ All formulas containing relevant symbols are relevant ◮ Iterate for n steps or up to fixpoint

Result: Unsatisfactory

◮ Some successes ◮ But: Selection to coarse ◮ Typically 5-7 iterations to fixpoint ◮ Often fixpoint is whole specification

14

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Stephan Schulz

Generalized SInE

Basic d-relation:

◮ Generality measure g : sig → N ◮ Formula C is in d-relation to f if f is g−minimal among all symbols in C

Algorithm:

◮ Start with the conjecture ◮ All symbols in the conjecture are relevant ◮ All formulas in d-relation with a relevant symbol are relevant ◮ Iterate up to fixpoint

15

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Stephan Schulz

Generalized SInE

Basic d-relation:

◮ Generality measure g : sig → N ◮ Formula C is in d-relation to f if f is g−minimal among all symbols in C

Algorithm:

◮ Start with the conjecture ◮ All symbols in the conjecture are relevant ◮ All formulas in d-relation with a relevant symbol are relevant ◮ Iterate up to fixpoint

Generalization:

◮ Different generality measures ◮ Different relaxation criteria for d-relations ◮ Limit iteration ◮ Limit number of formulas selected

15

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Stephan Schulz

First Results

Platform:

◮ 800 Mhz Quad-Core AMD Opteron ◮ 8GBytes RAM

CSR066+6 Plain E Server Mode Parsing Problem 109s Amortised Axiom Selection

  • 35 s

Clausification 240s 0.9 s Proof Timeout (>1200 s) 348 s

16

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Stephan Schulz

Evaluation: Large TPTP Problems

1 10 100 1000 10000 100000 1e+06 1e+07 2000 4000 6000 8000 10000 12000 14000 16000 TPTP 4.1.0 FOF and CNF problems Specification size (formulas)

17

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Stephan Schulz

Evaluation: Large TPTP Problems

1 10 100 1000 10000 100000 1e+06 1e+07 2000 4000 6000 8000 10000 12000 14000 16000 TPTP 4.1.0 FOF and CNF problems Specification size (formulas) 20000 axioms

17

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Stephan Schulz

Selected Results

TPTP 4.1.0 512 problems with ≥2000 axioms Platform:

◮ 2.2 GHz Intel Xeon ◮ 16 GBytes Strategy Solutions Full, 300 s 70 Full, 100 s 64 GSInE(f120,RUU,F100,L00020) 45 GSInE(t120,RUU,F100,L00050) 90 GSInE(f120,RUU,F100,L00100) 115 GSInE(t100,RUU,F100,L20000) 136 GSInE(t120,RUU,F100,L20000) 145 GSInE(t120,R10,F100,L20000) 148 GSInE(t120,RUU,F100,L00500) 151 GSInE(f120,RUU,F100,L01000) 154 GSInE(t120,RUU,F100,L05000) 157 GSInE(t150,RUU,F100,L20000) 159

18

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Stephan Schulz

All Results

20 40 60 80 100 120 140 160 180 300 100 gf120RUU_F100_L00020 gt120RUU_F100_L00020 gf120R01_F100_L20000 gt120R01_F100_L20000 gt120RUU_F100_L00050 gf120RUU_F100_L00050 gt120R02_F100_L20000 gf120R02_F100_L20000 gf120RUU_F100_L00100 gt120RUU_F100_L00100 gt120RUU_F100_L00200 gf120RUU_F100_L00200 gt100RUU_F100_L20000 gf110RUU_F100_L20000 gf100RUU_F100_L20000 gt110RUU_F100_L20000 gt120RUU_F100_L20000 gf120R10_F100_L20000 gf120R05_F100_L20000 gt120RUU_F100_L10000 gt120R10_F100_L20000 gf120RUU_F100_L00500 gf120RUU_F100_L10000 gt120R05_F100_L20000 gt120RUU_F100_L00500 gf120RUU_F100_L20000 gf120RUU_F100_L01000 gt120RUU_F100_L01000 gf120RUU_F100_L05000 gt120RUU_F100_L02000 gt120RUU_F100_L05000 gf120RUU_F100_L02000 gt150RUU_F100_L20000

19

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SLIDE 39

Stephan Schulz

Future Work

Complete server

◮ TCP connection ◮ Configurability ◮ Failure-tolerant parser

Implement client

◮ Telnet and netcat are sufficient but inconvenient

Evaluate different SInE variants

◮ Which paramters work for which problem clases? ◮ Which other generality measures are useful?

20

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SLIDE 40

Stephan Schulz CSR066-6: Smith et all, 2007

OpenCYC “Common sense” reasoning

  • 3341990 formulas
  • 480 MB of text (with

few comments)

Proof search

?

21

slide-41
SLIDE 41

Stephan Schulz CSR066-6: Smith et all, 2007

OpenCYC “Common sense” reasoning

  • 3341990 formulas
  • 480 MB of text (with

few comments)

Proof search

  • ~512000 steps
  • ~65 MB of text

21