A Mechanized Proof of Higman’s Lemma by Open Induction ⋆ Christian Sternagel University of Innsbruck, Austria January 18, 2016 Dagstuhl Seminar 16031 Well-Quasi-Orders in Computer Science ⋆ Supported by the Austrian Science Fund (FWF): P27502
Overview • Background • Higman’s Lemma by Open Induction • Conclusion C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 2/17
Background C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 3/17
Research Group Name: Computational Logic (headed by Aart Middeldorp) C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 4/17
Research Group Name: Computational Logic (headed by Aart Middeldorp) Main Research Topic Term Rewriting C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 4/17
Research Group Name: Computational Logic (headed by Aart Middeldorp) Main Research Topic Term Rewriting • termination, C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 4/17
Research Group Name: Computational Logic (headed by Aart Middeldorp) Main Research Topic Term Rewriting • termination, • confluence, C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 4/17
Research Group Name: Computational Logic (headed by Aart Middeldorp) Main Research Topic Term Rewriting • termination, • confluence, • and completion of term rewrite systems (TRSs) • . . . C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 4/17
Research Group Name: Computational Logic (headed by Aart Middeldorp) Main Research Topic Term Rewriting • termination, • confluence, • and completion of term rewrite systems (TRSs) • . . . • automated tools C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 4/17
Research Group Name: Computational Logic (headed by Aart Middeldorp) Main Research Topic Term Rewriting • termination, • confluence, • and completion of term rewrite systems (TRSs) • . . . • automated tools • certification C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 4/17
Automated Tools and Certification • (automatically) provide evidence TRS algorithms & techniques Literature Automated Tool Proof C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 5/17
Automated Tools and Certification • (automatically) provide evidence • (automatically) certify correctness of evidence TRS algorithms & techniques Literature Automated Tool theorems & proofs CPF Proof (XML) Proof Assistant code generation Formalization Certifier accept/reject C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 5/17
Automated Tools and Certification • (automatically) provide evidence • (automatically) certify correctness of evidence TRS algorithms & techniques Literature Automated Tool theorems & proofs Isabelle/HOL CPF Proof (XML) code generation IsaFoR Ce T A accept/reject C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 5/17
Demo • termination tool: T T T 2 • certifier: Ce T A C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 6/17
Higman’s Lemma by Open Induction C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 7/17
Bibliography Alfons Geser. A proof of Higman’s Lemma by open induction. Technical Report MIP-9606, Universit¨ at Passau, April 1996. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.35.8393 . Jean-Claude Raoult. Proving open properties by induction. Information Processing Letters , 29(1):19–23, 1988. doi: 10.1016/0020-0190(88)90126-3 . Mizuhito Ogawa and Christian Sternagel. Open Induction. Archive of Formal Proofs , November 2012. http://afp.sf.net/devel-entries/Open_Induction.shtml . C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 8/17
Higman’s Lemma Lemma: If set A is well-quasi-ordered then so is A ∗ . C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 9/17
Higman’s Lemma Lemma: If set A is well-quasi-ordered then so is A ∗ . Well-Quasi-Orders Definition: • a 1 , a 2 , a 3 , . . . ∈ A is ( ⊑ -)good if a i ⊑ a j for some i < j • ⊑ is almost-full (on A ) if all infinite ( A -)sequences are good • quasi-order ⊑ (on A ) is wqo (on A ) if ⊑ is almost-full (on A ) C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 9/17
Higman’s Lemma Lemma: If set A is well-quasi-ordered then so is A ∗ . Well-Quasi-Orders Definition: • a 1 , a 2 , a 3 , . . . ∈ A is ( ⊑ -)good if a i ⊑ a j for some i < j • ⊑ is almost-full (on A ) if all infinite ( A -)sequences are good • quasi-order ⊑ (on A ) is wqo (on A ) if ⊑ is almost-full (on A ) Nice Property: Every transitive extension � of almost-full ⊑ is well-founded. Proof. • assume a 1 ≻ a 2 ≻ a 3 ≻ . . . (with x ≻ y iff x � y and x �� y ) • by transitivity, a i ≻ a j for all i < j • then a i �⊑ a j for all i < j , and thus a is ⊑ -bad � C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 9/17
Higman’s Lemma Lemma: If ⊑ is wqo (on A ) then ⊑ ∗ is wqo (on A ∗ ). C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 10/17
Higman’s Lemma Lemma: If ⊑ is almost-full (on A ) then ⊑ ∗ is almost-full (on A ∗ ). C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 10/17
Higman’s Lemma Lemma: If ⊑ is almost-full (on A ) then ⊑ ∗ is almost-full (on A ∗ ). List Embedding Definition: embedding relation w.r.t. ⊑ : xs ⊑ ∗ ys xs ⊑ ∗ ys x ⊑ y xs ⊑ ∗ y · ys x · xs ⊑ ∗ y · ys [] ⊑ ∗ ys C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 10/17
Recall - Well-Founded Induction Schema: if ∀ x ∈ A. ( ∀ y ∈ A. y ≺ x − → P ( y )) − → P ( x ) then P ( x ) , for all x ∈ A , every well-founded ( A, ≺ ) and property P C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 11/17
Recall - Well-Founded Induction Schema: if ∀ x ∈ A. ( ∀ y ∈ A. y ≺ x − → P ( y )) − → P ( x ) then P ( x ) , for all x ∈ A , every well-founded ( A, ≺ ) and property P Generalization - Open Induction Theorem: if ∀ x ∈ A. ( ∀ y ∈ A. y ⊏ x − → P ( y )) − → P ( x ) then P ( x ) , for all x ∈ A , every downward complete quasi-order ( A, ⊑ ) and open property P C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 11/17
Recall - Well-Founded Induction Schema: if ∀ x ∈ A. ( ∀ y ∈ A. y ≺ x − → P ( y )) − → P ( x ) then P ( x ) , for all x ∈ A , every well-founded ( A, ≺ ) and property P Generalization - Open Induction Theorem: if ∀ x ∈ A. ( ∀ y ∈ A. y ⊏ x − → P ( y )) − → P ( x ) then P ( x ) , for all x ∈ A , every downward complete quasi-order ( A, ⊑ ) and open property P Definition: • ( A, ⊑ ) is downward complete if every non-empty ⊑ -chain C has a greatest lower bound (glb) g ∈ A . C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 11/17
Recall - Well-Founded Induction Schema: if ∀ x ∈ A. ( ∀ y ∈ A. y ≺ x − → P ( y )) − → P ( x ) then P ( x ) , for all x ∈ A , every well-founded ( A, ≺ ) and property P Generalization - Open Induction Theorem: if ∀ x ∈ A. ( ∀ y ∈ A. y ⊏ x − → P ( y )) − → P ( x ) then P ( x ) , for all x ∈ A , every downward complete quasi-order ( A, ⊑ ) and open property P Definition: • ( A, ⊑ ) is downward complete if every non-empty ⊑ -chain C has a greatest lower bound (glb) g ∈ A . • property P is ( ⊑ -)open if P ( g ) for some glb g implies P ( x ) for some x ∈ C , for every non-empty ⊑ -chain C C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 11/17
Lexicographic Order on Infinite Sequences Definition: a ≺ lex b iff a k ≺ b k and ∀ i < k. a i = b i for some k C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 12/17
Lexicographic Order on Infinite Sequences Definition: a ≺ lex b iff a k ≺ b k and ∀ i < k. a i = b i for some k Auxiliary Construction Definition: non-empty C and well-founded partial order (po) ≺ C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 12/17
Lexicographic Order on Infinite Sequences Definition: a ≺ lex b iff a k ≺ b k and ∀ i < k. a i = b i for some k Auxiliary Construction Definition: non-empty C and well-founded partial order (po) ≺ • E a k = { b ∈ C. ∀ i < k. a i = b i } C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 12/17
Lexicographic Order on Infinite Sequences Definition: a ≺ lex b iff a k ≺ b k and ∀ i < k. a i = b i for some k Auxiliary Construction Definition: non-empty C and well-founded partial order (po) ≺ • E a k = { b ∈ C. ∀ i < k. a i = b i } sequences from C equal to a up to k C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 12/17
Lexicographic Order on Infinite Sequences Definition: a ≺ lex b iff a k ≺ b k and ∀ i < k. a i = b i for some k Auxiliary Construction Definition: non-empty C and well-founded partial order (po) ≺ • E a k = { b ∈ C. ∀ i < k. a i = b i } • m i = min ≺ { a i | a ∈ E m i } C. Sternagel (University of Innsbruck) Dagstuhl Seminar 16031 12/17
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