Numeric termination metrics

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Sometimes, a proof proceeds not by a direct induction on some assumption, but by induction on some size function computed from an assumption. To mechanize such a proof in Twelf, you must make the size function explicit in the statement of the theorem.

This tutorial presents an example of such a proof. We show a fragment of a proof of confluence for a λ-calculus with typed η-expansion. The proof inducts on the size of a reduction derivation. Moreover, the proof uses %reduces to tell the termination checker that addends are subterms of their sum. In general, a %reduces declaration is necessary whenever the computation of a numeric termination metric uses an auxiliary relation like addition or maximum. See the tutorial on structural termination metrics for another approach to termination metrics.

Language Definition

The syntax, typing judgement, and reduction relation for the language are straightforward:

%% Syntax

tp  : type.  %name tp T.
o	: tp.
arrow	: tp -> tp -> tp.

exp : type.  %name exp E.
lam	: tp -> (exp -> exp) -> exp.
app	: exp -> exp -> exp.

%% Static Semantics

of : exp -> tp -> type.
of_lam	: of (lam T1 E) (arrow T1 T2)
	   <- ({x:exp} of x T1 -> of (E x) T2).
of_app	: of (app E1 E2) T2
	   <- of E1 (arrow T1 T2)
	   <- of E2 T1.

%% Dynamic Semantics

reduce : exp -> exp -> type.

reduce_id	: reduce E E.
reduce_lam	: reduce (lam T E) (lam T E')
		   <- ({x:exp} of x T -> reduce (E x) (E' x)).
reduce_app	: reduce (app E1 E2) (app E1' E2')
		   <- reduce E1 E1'
		   <- reduce E2 E2'.
reduce_beta	: reduce (app (lam T E1) E2) (E1' E2')
		   <- ({x:exp} of x T -> reduce (E1 x) (E1' x))
		   <- reduce E2 E2'.
reduce_eta	: reduce E (lam T1 ([x] app E' x))
		   <- of E (arrow T1 T2)
		   <- reduce E E'.

The judgement reduce defines a parallel, reflexive reduction relation with typed η-expansion.

Facts about numbers

In the proof below, we induct on the size of a reduction derivation. To get this induction to go through, we require some facts about addition on natural numbers.

First, we define addition:

nat : type.  %name nat N.

0 : nat.
s : nat -> nat.

1 : nat = s 0.

sum : nat -> nat -> nat -> type.

sum_0		: sum 0 N N.
sum_s		: sum (s N1) N2 (s N3)
		   <- sum N1 N2 N3.

For the proof below, we need a way to tell Twelf's termination checker that summands are subterms of their sum. We do that by proving a lemma with a %reduces declaration.

We prove the lemma for the second summand first. Note that all arguments of this lemma are inputs; the only "output" is the fact that the %reduces holds:

sum_reduces2 : {N1:nat} {N2:nat} {N3:nat} sum N1 N2 N3 -> type.
%mode sum_reduces2 +X1 +X2 +X3 +X4.

-: sum_reduces2 _ _ _ sum_0.
-: sum_reduces2 (s N1) N2 (s N3) (sum_s D)
    <- sum_reduces2 N1 N2 N3 D.

%worlds () (sum_reduces2 _ _ _ _).
%total D (sum_reduces2 _ _ _ D).
%reduces N2 <= N3 (sum_reduces2 N1 N2 N3 _).

The easiest way to prove the lemma for the first summand is to commute the addition and appeal to the previous lemma. We elide the proof of commutativity:

ERROR: option 'name' deprecated or invalid
sum_commute : sum N1 N2 N3 -> sum N2 N1 N3 -> type.
%mode sum_commute +D1 -D2.
%worlds () (sum_commute _ _).
sum_reduces1 : {N1:nat} {N2:nat} {N3:nat} sum N1 N2 N3 -> type.
%mode sum_reduces1 +X1 +X2 +X3 +X4.

-	: sum_reduces1 N1 N2 N3 Dsum
	   <- sum_commute Dsum Dsum'
	   <- sum_reduces2 N2 N1 N3 Dsum'.

%worlds () (sum_reduces1 _ _ _ _).
%total {} (sum_reduces1 _ _ _ _).
%reduces N1 <= N3 (sum_reduces1 N1 N2 N3 _).

Proof of confluence

We now show part of the proof of the diamond property for this notion of reduction. The proof requires a metric computing the size of a reduction derivation.

This article or section needs to recreate why you need a metric.

Definition of the metric

reduce_metric : reduce E E' -> nat -> type.

reduce_metric_id	: reduce_metric reduce_id 1.
reduce_metric_lam	: reduce_metric (reduce_lam D) (s N)
			   <- ({x:exp} {d:of x T} reduce_metric (D x d) N).
reduce_metric_app	: reduce_metric (reduce_app D2 D1) (s N)
			   <- reduce_metric D1 N1
			   <- reduce_metric D2 N2
			   <- sum N1 N2 N.
reduce_metric_beta	: reduce_metric (reduce_beta D2 D1) (s N)
			   <- ({x:exp} {d:of x T} reduce_metric (D1 x d) N1)
			   <- reduce_metric D2 N2
			   <- sum N1 N2 N.
reduce_metric_eta	: reduce_metric (reduce_eta D _) (s N)
			   <- reduce_metric D N.

Excerpt of the proof

diamond : {N1:nat} {N2:nat}
	   {D1:reduce E E1}
	   {D2:reduce E E2} 
	   reduce_metric D1 N1 
	   -> reduce_metric D2 N2
	   -> reduce E1 E'
	   -> reduce E2 E' -> type.
%mode diamond +X1 +X2 +X3 +X4 +X5 +X6 -X7 -X8.

-: diamond _ _ reduce_id D _ _ D reduce_id.
-: diamond _ _ D reduce_id _ _ reduce_id D.

-: diamond (s N1) (s N2)
    (reduce_lam D1) (reduce_lam D2) 
    (reduce_metric_lam DM1) (reduce_metric_lam DM2)
    (reduce_lam D1') (reduce_lam D2')
    <- ({x:exp} {d:of x T}
	  diamond N1 N2 (D1 x d) (D2 x d) (DM1 x d) (DM2 x d) (D1' x d) (D2' x d)).

-: diamond 
    (s N1)
    (s N2)
       (D21 : reduce E2 E21)
       (D11 : reduce E1 E11))
       (D22 : reduce E2 E22)
       (D12 : reduce E1 E12)) 
       (Dsum1 : sum N11 N21 N1)
       (DM21 : reduce_metric D21 N21)
       (DM11 : reduce_metric D11 N11))
       (Dsum2 : sum N12 N22 N2)
       (DM22 : reduce_metric D22 N22)
       (DM12 : reduce_metric D12 N12))
    (reduce_app D21' D11') (reduce_app D22' D12')
    <- sum_reduces1 N11 N21 N1 Dsum1
    <- sum_reduces2 N11 N21 N1 Dsum1
    <- sum_reduces1 N12 N22 N2 Dsum2
    <- sum_reduces2 N12 N22 N2 Dsum2
    <- diamond N11 N12 D11 D12 DM11 DM12 D11' D12'
    <- diamond N21 N22 D21 D22 DM21 DM22 D21' D22'.

%% fill in remaining cases

%block bind : some {t:tp} block {x:exp} {d:of x t}.
%worlds (bind) (diamond _ _ _ _ _ _ _ _).
%terminates [N1 N2] (diamond N1 N2 _ _ _ _ _ _).
See Twelf's output

The reduce_app against reduce_app case illustrates why we need to know that summands are subterms of their sum: the inductive calls are on the summands that add up to the size of the overall derivation. If we elided the calls to sum_reduces*, the case would not termination-check, because Twelf would not be able to tell that, for example, N11 < (s N1).

In other cases, which we have elided, the termination metric gets smaller but the reduction derivations themselves do not.

This article or section needs to show one such case..


We would like an overall theorem:

diamond/clean : reduce E E1
    	       -> reduce E E2
  	       -> reduce E1 E'
	       -> reduce E2 E' -> type.
%mode diamond/clean +X1 +X2 -X3 -X4.
%worlds (bind) (diamond/clean _ _ _ _ _ _ _ _).

It is simple to prove this theorem using the above if we prove an effectiveness lemma for reduce_metric.

Read more Twelf tutorials and other Twelf documentation.