[PPT] - A Mechanized Proof of Higmans Lemma by Open Induction Christian PowerPoint Presentation

SLIDE 1

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

SLIDE 2

Overview

Background
Higman’s Lemma by Open Induction
Conclusion
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 2/17

SLIDE 3

Background

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 3/17

SLIDE 4

Research Group

Name: Computational Logic (headed by Aart Middeldorp)

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 4/17

SLIDE 5

Research Group

Name: Computational Logic (headed by Aart Middeldorp)

Main Research Topic

Term Rewriting

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 4/17

SLIDE 6

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

SLIDE 7

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

SLIDE 8

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

SLIDE 9

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

SLIDE 10

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

SLIDE 11

Automated Tools and Certification

(automatically) provide evidence

Literature Automated Tool algorithms & techniques TRS Proof

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 5/17

SLIDE 12

Automated Tools and Certification

(automatically) provide evidence
(automatically) certify correctness of evidence

Literature Automated Tool algorithms & techniques TRS Proof CPF (XML) Proof Assistant Formalization Certifier theorems & proofs code generation accept/reject

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 5/17

SLIDE 13

Automated Tools and Certification

(automatically) provide evidence
(automatically) certify correctness of evidence

Literature Automated Tool algorithms & techniques TRS Proof CPF (XML) Isabelle/HOL IsaFoR Ce T A theorems & proofs code generation accept/reject

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 5/17

SLIDE 14

Demo

termination tool: T

T T 2

certifier: Ce

T A

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 6/17

SLIDE 15

Higman’s Lemma by Open Induction

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 7/17

SLIDE 16

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

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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

SLIDE 18

Higman’s Lemma

Lemma: If set A is well-quasi-ordered then so is A∗.

Well-Quasi-Orders

Definition:

a1, a2, a3, . . . ∈ A is (⊑-)good if ai ⊑ aj 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

SLIDE 19

Higman’s Lemma

Lemma: If set A is well-quasi-ordered then so is A∗.

Well-Quasi-Orders

Definition:

a1, a2, a3, . . . ∈ A is (⊑-)good if ai ⊑ aj 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 a1 ≻ a2 ≻ a3 ≻ . . . (with x ≻ y iff x y and x y)
by transitivity, ai ≻ aj for all i < j
then ai ⊑ aj for all i < j, and thus a is ⊑-bad
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 9/17

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Higman’s Lemma

Lemma: If ⊑ is wqo (on A) then ⊑∗ is wqo (on A∗).

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 10/17

SLIDE 21

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

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Higman’s Lemma

Lemma: If ⊑ is almost-full (on A) then ⊑∗ is almost-full (on A∗).

List Embedding

Definition: embedding relation w.r.t. ⊑: [] ⊑∗ ys xs ⊑∗ ys xs ⊑∗ y · ys x ⊑ y xs ⊑∗ ys x · xs ⊑∗ y · ys

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 10/17

SLIDE 23

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

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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

SLIDE 25

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

SLIDE 26

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

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Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 28

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi 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

SLIDE 29

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 30

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

sequences from C equal to a up to k

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 31

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 32

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

{ai | a ∈ A} is i-th “column” of A

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 33

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

min≺A is some element

f A s.t. no other x ∈ A

is ≺-smaller

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 34

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

Example

3 7 9 5 8 4 6 5 8 . . . 7 4 2 7 4 4 6 7 9 . . . 1 4 4 2 9 2 2 1 8 . . . 1 4 5 3 8 6 8 8 6 . . . 1 4 4 7 7 4 3 7 1 . . . 7 1 7 8 4 6 9 1 6 . . .

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 35

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

Example

3 7 9 5 8 4 6 5 8 . . . 7 4 2 7 4 4 6 7 9 . . . 1 4 4 2 9 2 2 1 8 . . . 1 4 5 3 8 6 8 8 6 . . . 1 4 4 7 7 4 3 7 1 . . . 7 1 7 8 4 6 9 1 6 . . .

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 36

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

Example

3 7 9 5 8 4 6 5 8 . . . 7 4 2 7 4 4 6 7 9 . . . 1 4 4 2 9 2 2 1 8 . . . 1 4 5 3 8 6 8 8 6 . . . 1 4 4 7 7 4 3 7 1 . . . 7 1 7 8 4 6 9 1 6 . . .

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 37

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

Example

3 7 9 5 8 4 6 5 8 . . . 7 4 2 7 4 4 6 7 9 . . . 1 4 4 2 9 2 2 1 8 . . . 1 4 5 3 8 6 8 8 6 . . . 1 4 4 7 7 4 3 7 1 . . . 7 1 7 8 4 6 9 1 6 . . .

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 38

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

Example

3 7 9 5 8 4 6 5 8 . . . 7 4 2 7 4 4 6 7 9 . . . 1 4 4 2 9 2 2 1 8 . . . 1 4 5 3 8 6 8 8 6 . . . 1 4 4 7 7 4 3 7 1 . . . 7 1 7 8 4 6 9 1 6 . . .

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 39

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

Example

3 7 9 5 8 4 6 5 8 . . . 7 4 2 7 4 4 6 7 9 . . . 1 4 4 2 9 2 2 1 8 . . . 1 4 5 3 8 6 8 8 6 . . . 1 4 4 7 7 4 3 7 1 . . . 7 1 7 8 4 6 9 1 6 . . .

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 40

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

Example

3 7 9 5 8 4 6 5 8 . . . 7 4 2 7 4 4 6 7 9 . . . 1 4 4 2 9 2 2 1 8 . . . 1 4 5 3 8 6 8 8 6 . . . 1 4 4 7 7 4 3 7 1 . . . 7 1 7 8 4 6 9 1 6 . . .

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 41

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

Example

3 7 9 5 8 4 6 5 8 . . . 7 4 2 7 4 4 6 7 9 . . . 1 4 4 2 9 2 2 1 8 . . . 1 4 5 3 8 6 8 8 6 . . . 1 4 4 7 7 4 3 7 1 . . . 7 1 7 8 4 6 9 1 6 . . .

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 42

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

Example

3 7 9 5 8 4 6 5 8 . . . 7 4 2 7 4 4 6 7 9 . . . 1 4 4 2 9 2 2 1 8 . . . 1 4 5 3 8 6 8 8 6 . . . 1 4 4 7 7 4 3 7 1 . . . 7 1 7 8 4 6 9 1 6 . . .

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 43

Lexicographic Order on Infinite Sequences

Definition: a ≺lex b iff ak ≺ bk and ∀i < k. ai = bi for some k

Auxiliary Construction

Definition: non-empty C and well-founded partial order (po) ≺

Ea

k = {b ∈ C. ∀i < k. ai = bi}

mi = min≺{ai | a ∈ Em

i }

Example

3 7 9 5 8 4 6 5 8 . . . 7 4 2 7 4 4 6 7 9 . . . 1 4 4 2 9 2 2 1 8 . . . 1 4 5 3 8 6 8 8 6 . . . 1 4 4 7 7 4 3 7 1 . . . 7 1 7 8 4 6 9 1 6 . . .

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 12/17

SLIDE 44

Lexicographic Order on Infinite Sequences (cont’d)

Lemma: m is lower bound of any non-empty ≺lex-chain C Proof.

assume a ∈ C and a = m
take least k s.t. ak = mk (thus ∀i < k. ai = mi)
then a ∈ Em

k and hence ak ∈ {bk | b ∈ Em k }

thus mk ≺ ak since ak ≺ mk and C is ≺lex-chain
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 13/17

SLIDE 45

Lexicographic Order on Infinite Sequences (cont’d)

Lemma: m is lower bound of any non-empty ≺lex-chain C Proof.

assume a ∈ C and a = m
take least k s.t. ak = mk (thus ∀i < k. ai = mi)
then a ∈ Em

k and hence ak ∈ {bk | b ∈ Em k }

thus mk ≺ ak since ak ≺ mk and C is ≺lex-chain

Lemma: m is glb of any non-empty ≺lex-chain C Proof.

assume ℓ is lower bound and ℓ = m
take least k s.t. ℓk = mk (thus ∀i < k. ℓi = mi)
obtain a ∈ Em

k+1 (i.e., ∀i ≤ k. ai = mi)

then ℓk ≺ ak (since ℓ is lb) and thus ℓ ≺lex m
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 13/17

SLIDE 46

Lexicographic Order on Infinite Sequences (cont’d)

Lemma: m is lower bound of any non-empty ≺lex-chain C Proof.

assume a ∈ C and a = m
take least k s.t. ak = mk (thus ∀i < k. ai = mi)
then a ∈ Em

k and hence ak ∈ {bk | b ∈ Em k }

thus mk ≺ ak since ak ≺ mk and C is ≺lex-chain

Lemma: m is glb of any non-empty ≺lex-chain C Proof.

assume ℓ is lower bound and ℓ = m
take least k s.t. ℓk = mk (thus ∀i < k. ℓi = mi)
obtain a ∈ Em

k+1 (i.e., ∀i ≤ k. ai = mi)

then ℓk ≺ ak (since ℓ is lb) and thus ℓ ≺lex m

Corollary: ≺lex is downward complete for every well-founded po ≺

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 13/17

SLIDE 47

Lexicographic Order on Infinite Sequences (cont’d)

Lemma: being ⊑-good is an open property Proof.

assume C is non-empty ≺lex-chain with ⊑-good glb g
then m is ⊑-good (since g = m)
thus mi ⊑ mj for some i < j
moreover a ∈ Em

j+1 for some a

but then ai = mi and aj = mj and thus ai ⊑ aj
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 14/17

SLIDE 48

Proof of Higman’s Lemma

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 49

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 50

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 51

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po

x ⊳ y iff y = w @ x for some w = []

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 52

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
let a1, a2, a3, . . . ∈ A, show a is ⊑∗-good by open induction
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 53

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
let a1, a2, a3, . . . ∈ A, show a is ⊑∗-good by open induction
IH: any sequence b1, b2, b3, . . . ∈ A s.t. b ⊳lex a is ⊑∗-good
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 54

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
let a1, a2, a3, . . . ∈ A, show a is ⊑∗-good by open induction
IH: any sequence b1, b2, b3, . . . ∈ A s.t. b ⊳lex a is ⊑∗-good
note ai = hi · ti for all i ≥ 1 (otherwise a is trivially good)
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 55

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
let a1, a2, a3, . . . ∈ A, show a is ⊑∗-good by open induction
IH: any sequence b1, b2, b3, . . . ∈ A s.t. b ⊳lex a is ⊑∗-good
note ai = hi · ti for all i ≥ 1 (otherwise a is trivially good)
obtain hσ(1) ⊑ hσ(2) ⊑ hσ(3) ⊑ · · · (since ⊑ is almost-full)
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 56

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
let a1, a2, a3, . . . ∈ A, show a is ⊑∗-good by open induction
IH: any sequence b1, b2, b3, . . . ∈ A s.t. b ⊳lex a is ⊑∗-good
note ai = hi · ti for all i ≥ 1 (otherwise a is trivially good)
obtain hσ(1) ⊑ hσ(2) ⊑ hσ(3) ⊑ · · · (since ⊑ is almost-full)
let a′ = a1, a2, . . . , aσ(1)−1, tσ(1), tσ(2), . . .
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 57

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
let a1, a2, a3, . . . ∈ A, show a is ⊑∗-good by open induction
IH: any sequence b1, b2, b3, . . . ∈ A s.t. b ⊳lex a is ⊑∗-good
note ai = hi · ti for all i ≥ 1 (otherwise a is trivially good)
obtain hσ(1) ⊑ hσ(2) ⊑ hσ(3) ⊑ · · · (since ⊑ is almost-full)
let a′ = a1, a2, . . . , aσ(1)−1, tσ(1), tσ(2), . . .
then a′ ⊳lex a and thus a′

i ⊑∗ a′ j for some i < j by IH

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 58

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
let a1, a2, a3, . . . ∈ A, show a is ⊑∗-good by open induction
IH: any sequence b1, b2, b3, . . . ∈ A s.t. b ⊳lex a is ⊑∗-good
note ai = hi · ti for all i ≥ 1 (otherwise a is trivially good)
obtain hσ(1) ⊑ hσ(2) ⊑ hσ(3) ⊑ · · · (since ⊑ is almost-full)
let a′ = a1, a2, . . . , aσ(1)−1, tσ(1), tσ(2), . . .
then a′ ⊳lex a and thus a′

i ⊑∗ a′ j for some i < j by IH

a1 a2 · · · aσ(1)−1 tσ(1) tσ(2) · · ·

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 59

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
let a1, a2, a3, . . . ∈ A, show a is ⊑∗-good by open induction
IH: any sequence b1, b2, b3, . . . ∈ A s.t. b ⊳lex a is ⊑∗-good
note ai = hi · ti for all i ≥ 1 (otherwise a is trivially good)
obtain hσ(1) ⊑ hσ(2) ⊑ hσ(3) ⊑ · · · (since ⊑ is almost-full)
let a′ = a1, a2, . . . , aσ(1)−1, tσ(1), tσ(2), . . .
then a′ ⊳lex a and thus a′

i ⊑∗ a′ j for some i < j by IH

a1 a2 · · · aσ(1)−1 tσ(1) tσ(2) · · ·

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 60

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
let a1, a2, a3, . . . ∈ A, show a is ⊑∗-good by open induction
IH: any sequence b1, b2, b3, . . . ∈ A s.t. b ⊳lex a is ⊑∗-good
note ai = hi · ti for all i ≥ 1 (otherwise a is trivially good)
obtain hσ(1) ⊑ hσ(2) ⊑ hσ(3) ⊑ · · · (since ⊑ is almost-full)
let a′ = a1, a2, . . . , aσ(1)−1, tσ(1), tσ(2), . . .
then a′ ⊳lex a and thus a′

i ⊑∗ a′ j for some i < j by IH

a1 a2 · · · aσ(1)−1 tσ(1) tσ(2) · · · ai ⊑∗ aj case 1

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 61

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
let a1, a2, a3, . . . ∈ A, show a is ⊑∗-good by open induction
IH: any sequence b1, b2, b3, . . . ∈ A s.t. b ⊳lex a is ⊑∗-good
note ai = hi · ti for all i ≥ 1 (otherwise a is trivially good)
obtain hσ(1) ⊑ hσ(2) ⊑ hσ(3) ⊑ · · · (since ⊑ is almost-full)
let a′ = a1, a2, . . . , aσ(1)−1, tσ(1), tσ(2), . . .
then a′ ⊳lex a and thus a′

i ⊑∗ a′ j for some i < j by IH

a1 a2 · · · aσ(1)−1 tσ(1) tσ(2) · · · ai ⊑∗ tσ(j′) case 2

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 62

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
let a1, a2, a3, . . . ∈ A, show a is ⊑∗-good by open induction
IH: any sequence b1, b2, b3, . . . ∈ A s.t. b ⊳lex a is ⊑∗-good
note ai = hi · ti for all i ≥ 1 (otherwise a is trivially good)
obtain hσ(1) ⊑ hσ(2) ⊑ hσ(3) ⊑ · · · (since ⊑ is almost-full)
let a′ = a1, a2, . . . , aσ(1)−1, tσ(1), tσ(2), . . .
then a′ ⊳lex a and thus a′

i ⊑∗ a′ j for some i < j by IH

a1 a2 · · · aσ(1)−1 tσ(1) tσ(2) · · · tσ(i′) ⊑∗ tσ(j′) case 3

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 63

Proof of Higman’s Lemma

assume ⊑ is almost-full on A
note that suffix relation ⊳ is well-founded po
let a1, a2, a3, . . . ∈ A, show a is ⊑∗-good by open induction
IH: any sequence b1, b2, b3, . . . ∈ A s.t. b ⊳lex a is ⊑∗-good
note ai = hi · ti for all i ≥ 1 (otherwise a is trivially good)
obtain hσ(1) ⊑ hσ(2) ⊑ hσ(3) ⊑ · · · (since ⊑ is almost-full)
let a′ = a1, a2, . . . , aσ(1)−1, tσ(1), tσ(2), . . .
then a′ ⊳lex a and thus a′

i ⊑∗ a′ j for some i < j by IH

a1 a2 · · · aσ(1)−1 tσ(1) tσ(2) · · ·

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 15/17

SLIDE 64

Conclusion

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 16/17

SLIDE 65

Questions

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 17/17

SLIDE 66

Questions

why reprove already formalized results? realizability?
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 17/17

SLIDE 67

Questions

why reprove already formalized results? realizability?
what about realizability of “Ramsey”-step in above proof (i.e.,

hσ(1) ⊑ hσ(2) ⊑ hσ(3) ⊑ · · · )

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 17/17

SLIDE 68

Questions

why reprove already formalized results? realizability?
what about realizability of “Ramsey”-step in above proof (i.e.,

hσ(1) ⊑ hσ(2) ⊑ hσ(3) ⊑ · · · )

prove Kruskal by open induction?
C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 17/17

SLIDE 69

Questions

why reprove already formalized results? realizability?
what about realizability of “Ramsey”-step in above proof (i.e.,

hσ(1) ⊑ hσ(2) ⊑ hσ(3) ⊑ · · · )

prove Kruskal by open induction?
suitable notion of “simplification order” for reduction pairs

(fact: none of the well-foundedness proofs inside IsaFoR is based on Kruskal right now)

C. Sternagel (University of Innsbruck)

Dagstuhl Seminar 16031 17/17