Constructivization. LFMTP June CNAM, Inria, LSV. A Rewrite System for Proof. Constructivization. Raphaël Cauderlier.
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1 CNAM, Inria, LSV LFMTP June / 36
2 Discharging proofs to a theorem prover Theorem provers are classical Proof assistants for type theory are constructive Assume the classical axiom no proof-as-program Restrict to the image of a translation too small In generated proofs, classical axioms are always used but not always necessary. 2 / 36
3 3 / 36
4 Problem Given a formula ϕ and a classical proof π of ϕ, is ϕ constructively provable? 4 / 36
5 Problem Given a formula ϕ and a classical proof π of ϕ, is ϕ constructively provable? As hard as constructive provability (ϕ := ψ P P) 4 / 36
6 Problem Given a formula ϕ and a classical proof π of ϕ, is ϕ constructively provable? As hard as constructive provability (ϕ := ψ P P) Heuristic approach 4 / 36
7 λπ-calculus modulo Extends λπ-calculus with rewriting: Signature Σ contains (well-typed) R Conversion is extended Σ; Γ t : A Σ; Γ B : Type (when A Rβ B) Σ; Γ t : B 5 / 36
8 Dedukti: a logical framework based on rewriting Universal proof checker interactive and automatic provers Purely functional programming language based on rewriting Proofs are objects, we can compute on them beyond cut elimination. 6 / 36
9 Syntax of First-Order Logic A ::= P A A A A A A x.a(x) x.a(x) A := A 7 / 36
10 Classical axioms Classical logic = Intuitionistic logic + any one of those: Law of Double Negation: dn A : A A Law of Excluded Middle: em A : A A... 8 / 36
11 Proof constructivization Dedukti world Library of first-order problems Theorem prover First-order problem constructivization (Constructive?) proof Classical proof 9 / 36
12 A rewrite system for A A The following statements are constructive theorems: t : EM t : EM t : (EM A EM B ) EM A B t : (EM A EM B ) EM A B t : (EM A EM B ) EM A B where EM A := A A which leads to the following partial definition of em: 10 / 36
13 A rewrite system for A A EM A := A A em A : EM A em t em t em A B t (em A, em B ) em A B t (em A, em B ) em A B t (em A, em B ) 11 / 36
14 A rewrite system for A A Nothing smart to do with quantifiers and atoms In practice, never yields a constructive proof We do not inspect the proof 12 / 36
15 Example: A A dn A A (λh (A A).H (A A) (λh A.E A ((λh A.H (A A)(λH A.H A ))H A ))) 13 / 36
16 Example: A A dn A A (λh (A A).H (A A) (λh A.E A ((λh A.H (A A)(λH A.H A ))H A ))) dn A B d (dn A, dn B ) 14 / 36
17 Example: A A λh A.dn A (λh A. (λh (A A).H (A A) (λh A.E A((λH A.H (A A)(λH (λh A A.H A (H A A H A )) A.H A ))H A ))) 15 / 36
18 Example: A A λh A.dn A (λh A. (λh (A A).H (A A) (λh A.E A((λH A.H (A A)(λH (λh A A.H A (H A A H A )) A.H A ))H A ))) E A (π A π A ) π A 16 / 36
19 Example: A A λh A.dn A (λh A. (λh (A A).H (A A) (λh A.H A )) (λh A A.H A (H A A H A )) 17 / 36
20 Example: A A λh A.dn A (λh A. (λh (A A).H (A A) (λh A.H A )) (λh A A.H A (H A A H A )) (λh A.π B (H A )) π A π B (π A ) 18 / 36
21 Example: A A λh A.dn A (λh A. (λh A A.H A (H A A H A )) (λh A.H A )) 19 / 36
22 Example: A A λh A.dn A (λh A. (λh A A.H A (H A A H A )) (λh A.H A )) (λh A.π B (H A )) π A π B (π A ) 20 / 36
23 Example: A A λh A.dn A (λh A. H A ((λh A.H A ) H A)) 21 / 36
24 Example: A A λh A.dn A (λh A. H A ((λh A.H A ) H A)) (λh A.π B (H A )) π A π B (π A ) 22 / 36
25 Example: A A λh A.dn A (λh A. H A H A ) 23 / 36
26 Example: A A λh A.dn A (λh A. H A H A ) dn A (λh A.H A π A ) π A 24 / 36
27 Example: A A λh A.H A 25 / 36
28 A rewrite system for A A The following statements are constructive theorems: d : DN d : DN d : (DN A DN B ) DN A B d : DN B DN A B d : ( x.dn P(x) ) DN x.p(x) where DN A := A A which leads to the following partial definition of dn: 26 / 36
29 A rewrite system for A A DN A := A A dn A : DN A dn d dn d dn A B d (dn A, dn B ) dn A B d (dn B ) dn x.p(x) d (λx.dn P(x) ) 27 / 36
30 A rewrite system for A A DN A := A A dn A : DN A dn d dn d dn A B d (dn A, dn B ) dn A B d (dn B ) dn x.p(x) d (λx.dn P(x) ) dn A (λh A.H A π A ) π A 27 / 36
31 A rewrite system for A A DN A := A A dn A : DN A dn d dn d dn A B d (dn A, dn B ) dn A B d (dn B ) dn x.p(x) d (λx.dn P(x) ) dn A (λh A.H A π A ) π A dn A (λ.π ) E A (π ) 27 / 36
32 Elimination of negation proofs Higher-order dn A (λh A.H A π A ) π A dn A (λ.π ) E A (π ) 28 / 36
33 Elimination of negation proofs Higher-order dn A (λh A.H A π A ) π A dn A (λ.π ) E A (π ) E A (π A π A ) π A 28 / 36
34 Elimination of negation proofs Higher-order dn A (λh A.H A π A ) π A dn A (λ.π ) E A (π ) E A(π A π A ) π A E A B (π ) π A E B(π ) λh A.E B (π ) E A B (π ) 28 / 36
35 Confluence λh.h λh.dn (λh.h H ) λh.i 29 / 36
36 The constructivizatoin pipeline Dedukti world Library of first-order problems Theorem prover First-order problem constructivization (Constructive?) proof Classical proof 30 / 36
37 Zooming at the constructivization box First-order problem Proof with dn dn (λ.π A ) E A (π ) dn (λh A A.H A π A ) π A dn... dn... dn A B... dn A B... dn x.a(x)... dn A (λh A.E )... dn A (λh A.E )... E A (π A π A ) π A λh A.E B (π ) E A B (π ) E A B (π ) π A E B (π ) (Constructive?) proof dn... (...)... constructivization 31 / 36
38 Meta-Dedukti Dedukti is good at writing Dedukti code. Normalize the same term with respect to several successive rewrite Rewrite system = meta-programming stage Confluence of the last 32 / 36
39 Results Timeout: 10s (Zenon) and 10min (Dedukti) Memory limit: 2GB Library: TPTP v6.3.0 Problems 6528 Classical proofs 1258 Normalized proofs 1240 Constructive proofs 856 rate 68% 33 / 36
40 Related work Zenonide in sequent calculus Specific to Zenon Similar performances (66.8%) Complementary (76% combined) ileancop Best intuitionistic theorem prover according to ILTP No proof output More ambitious: complete for first-order intuitionistic logic Surprise: we are competitive 34 / 36
41 Simple heuristics for proof constructivization Dedukti as a meta-language for transforming proofs Constructivizators can be chained Most classical proofs generated by Zenon are constructive Theorems = total functions, Axioms = partial functions 35 / 36
42 Future work Syntax for the meta-language Try other provers (iprover Modulo, VeriT) Deduction modulo Higher-order Intermediate logics Other axioms (extensionality, univalence, choice) 36 / 36
43 Future work Syntax for the meta-language Try other provers (iprover Modulo, VeriT) Deduction modulo Higher-order Intermediate logics Other axioms (extensionality, univalence, choice) Thank you for your attention! 36 / 36
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