Original: Therefore, the closure of F is,
Revised: Therefore, the closure of F is,

Original: [Equation]
Revised: [Equation]

Original: with
Revised: with

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: To get the desired result, we need to get rid of the condition “ f ( x ) ∈ F ( x ) ” in the description of F .
Revised: To get the desired result, we need to get rid of the condition “ f ( x ) ∈ F ( x ) ” in the description of F .

Original: Fix x ∈ X .
Revised: Fix x ∈ X .

Original: First, we show that
Revised: First, we show that

Original: [Equation]
Revised: [Equation]

Original: with
Revised: with

Original: [Equation]
Revised: [Equation]

Original: (note that the equalities here are not in R anymore but in F ).
Revised: (note that the equalities here are not in R anymore but in F ).

Original: The direct inclusion “ ⊆ ” is immediate by construction of J ( x ) so we focus on the reverse direction.
Revised: The direct inclusion “ ⊆ ” is immediate by construction of J ( x ) so we focus on the reverse direction.

Original: For this, we show that the 4-tuples ( i, i (cid:48) , j, j (cid:48) ) of I ( x ) necessarily satisfy i (cid:48) =
Revised: For this, we show that the 4-tuples ( i, i (cid:48) , j, j (cid:48) ) of I ( x ) necessarily satisfy i (cid:48) =

Original: j (cid:48) and ( i, j
Revised: j (cid:48) and ( i, j

Original: ) ∈ J ( x ) .
Revised: ) ∈ J ( x ) .

Original: First note that, by the ﬁrst assumption, the vector y 0 ∈ F n such that y 

Revised: First note that, by the ﬁrst assumption, the vector y 0 ∈ F n such that y

Original: i
Revised: i

Original: =
Revised: =

Original: . , p ) for i ∈ { 1 , . .
Revised: . , p ) for i ∈ { 1 , . .

Original: . , n } is in F ( x ) .
Revised: . , n } is in F ( x ) .

Original: Indeed, take the constant function always equal to (1 , 2 , . . .
Revised: Indeed, take the constant function always equal to (1 , 2 , . . .

Original: , p ) as h .
Revised: , p ) as h .

Original: Now, consider a 4-tuple ( i, i (cid:48) , j, j (cid:48) ) of I ( x ) and we show that, actually, ( i, j ) ∈ J ( x ) and i (cid:48)
Revised: Now, consider a 4-tuple ( i, i (cid:48) , j, j (cid:48) ) of I ( x ) and we show that, actually, ( i, j ) ∈ J ( x ) and i (cid:48)

Original: =
Revised: =

Original: j (cid:48) .
Revised: j (cid:48) .

Original: As yin F ( x ) , and y 

Revised: As yin F ( x ) , and y

Original: i,i (cid:48)
Revised: i,i (cid:48)

Original: =
Revised: =

Original: i (cid:48) , y 0 j,j (cid:48) =
Revised: i (cid:48) , y 0 j,j (cid:48) =

Original: j (cid:48) , ( i, i (cid:48) , j, j (cid:48) )
Revised: j (cid:48) , ( i, i (cid:48) , j, j (cid:48) )

Original: ∈
Revised: ∈

Original: I ( x ) implies that j (cid:48) =
Revised: I ( x ) implies that j (cid:48) =

Original: i (cid:48) .
Revised: i (cid:48) .

Original: Consider, k ∈ { 1 , . . .
Revised: Consider, k ∈ { 1 , . . .

Original: , p } .
Revised: , p } .

Original: We show that, for any y ∈ F ( x ) , y i,k = y j,k .
Revised: We show that, for any y ∈ F ( x ) , y i,k = y j,k .

Original: But such a y can be written as y = f
Revised: But such a y can be written as y = f

Original: ( x ) for some f ∈ F .
Revised: ( x ) for some f ∈ F .

Original: Consider, the function h ∈ C
Revised: Consider, the function h ∈ C

Original: ( F , F ) associated to the permutation ( i (cid:48) k ) , deﬁned by, z (cid:55)→ ( z ( i (cid:48) k )(1) , . . .
Revised: ( F , F ) associated to the permutation ( i (cid:48) k ) , deﬁned by, z (cid:55)→ ( z ( i (cid:48) k )(1) , . . .

Original: , z ( i (cid:48) k )( n ) ) =
Revised: , z ( i (cid:48) k )( n ) ) =

Original: ( z 1 , . . .
Revised: ( z 1 , . . .

Original: , z i (cid:48) − 1 , z k , z i (cid:48) +1 , . . . , z k − 1 , z i (cid:48) , z k +1 , . . .
Revised: , z i (cid:48) − 1 , z k , z i (cid:48) +1 , . . . , z k − 1 , z i (cid:48) , z k +1 , . . .

Original: , z p ) .
Revised: , z p ) .

Original: By our ﬁrst assumption, z (cid:55)→ ( h ( f ( z ) 1 ) , . . .
Revised: By our ﬁrst assumption, z (cid:55)→ ( h ( f ( z ) 1 ) , . . .

Original: , h ( f ( z ) n ))
Revised: , h ( f ( z ) n ))

Original: ∈ F so that ( h ( y 1 ) , . .
Revised: ∈ F so that ( h ( y 1 ) , . .

Original: , h ( y n ))
Revised: , h ( y n ))

Original: ∈ F ( x ) .
Revised: ∈ F ( x ) .

Original: In particular, as ( i, i (cid:48) , j, i (cid:48) ) ∈
Revised: In particular, as ( i, i (cid:48) , j, i (cid:48) ) ∈

Original: I ( x ) , h ( y i ) i (cid:48)
Revised: I ( x ) , h ( y i ) i (cid:48)

Original: =
Revised: =

Original: h ( y j ) i (cid:48) , i.e. y i,k = y j,k .
Revised: h ( y j ) i (cid:48) , i.e. y i,k = y j,k .

Original: Therefore, ( i, j ) ∈ J ( x ) .
Revised: Therefore, ( i, j ) ∈ J ( x ) .

Original: Finally, we can conclude that F ( x ) ⊃ { y ∈ F n : ∀ ( i, j ) ∈ J ( x ) , y i = y j } .
Revised: Finally, we can conclude that F ( x ) ⊃ { y ∈ F n : ∀ ( i, j ) ∈ J ( x ) , y i = y j } .

Original: Indeed, take y ∈ F n : ∀ ( i, j
Revised: Indeed, take y ∈ F n : ∀ ( i, j

Original: ) ∈ J ( x ) , y i =
Revised: ) ∈ J ( x ) , y i =

Original: y j .
Revised: y j .

Original: We show that all the constraints of I ( x ) are satisﬁed.
Revised: We show that all the constraints of I ( x ) are satisﬁed.

Original: Indeed, take ( i, i (cid:48) , j, j (cid:48) )
Revised: Indeed, take ( i, i (cid:48) , j, j (cid:48) )

Original: ∈
Revised: ∈

Original: I ( x ) .
Revised: I ( x ) .

Original: We have shown that ( i, j ) ∈
Revised: We have shown that ( i, j ) ∈

Original: J ( x ) and i (cid:48) =
Revised: J ( x ) and i (cid:48) =

Original: j (cid:48) so that y i =
Revised: j (cid:48) so that y i =

Original: y j and in particular
Revised: y j and in particular

Original: y i,i (cid:48) =
Revised: y i,i (cid:48) =

Original: y j,j (cid:48) .
Revised: y j,j (cid:48) .

Original: Therefore, this ﬁnishes the proof of F ( x ) ⊃ { y ∈ F n : ∀ ( i, j
Revised: Therefore, this ﬁnishes the proof of F ( x ) ⊃ { y ∈ F n : ∀ ( i, j

Original: ) ∈ J ( x ) , y i = y j } .
Revised: ) ∈ J ( x ) , y i = y j } .

Original: Thus,
Revised: Thus,

Original: F ( x ) = { y ∈ F n : ∀ ( i, j ) ∈ J ( x ) , y i =
Revised: F ( x ) = { y ∈ F n : ∀ ( i, j ) ∈ J ( x ) , y i =

Original: y j } , with
Revised: y j } , with

Original: J ( x ) =
Revised: J ( x ) =

Original: { ( i, j ) ∈ { 1 , . . . , n } 2 : y i =
Revised: { ( i, j ) ∈ { 1 , . . . , n } 2 : y i =

Original: y j } .
Revised: y j } .

Original: We have proven so far, that,
Revised: We have proven so far, that,

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation]
Revised: [Equation]

Original: Take h ∈ C E ( X, F n ) such that ρ ( F ) ⊆
Revised: Take h ∈ C E ( X, F n ) such that ρ ( F ) ⊆

Original: ρ ( h ) and ﬁx x ∈ X .
Revised: ρ ( h ) and ﬁx x ∈ X .

Original: Our goal is to show that, for any ( i, j ) ∈ J ( x ) , h ( x ) i
Revised: Our goal is to show that, for any ( i, j ) ∈ J ( x ) , h ( x ) i

Original: = h ( x ) j so that h ( x ) ∈ F ( x ) .
Revised: = h ( x ) j so that h ( x ) ∈ F ( x ) .

Original: But, if ( i, j ) ∈ J ( x ) , then for any f ∈ F , f ( x ) i =
Revised: But, if ( i, j ) ∈ J ( x ) , then for any f ∈ F , f ( x ) i =

Original: f ( x ) j so that ( i j ) · f ( x ) =
Revised: f ( x ) j so that ( i j ) · f ( x ) =

Original: f
Revised: f

Original: ( x ) , where ( i j ) denotes the permutation which exchanges i and j .
Revised: ( x ) , where ( i j ) denotes the permutation which exchanges i and j .

Original: Moreover, as ( i j ) ∈ S n , by equivariance, this means that f (( i j ) · x ) =
Revised: Moreover, as ( i j ) ∈ S n , by equivariance, this means that f (( i j ) · x ) =

Original: f
Revised: f

Original: ( x ) for every f ∈ F and therefore that (( i j ) · x, x ) ∈ ρ ( F ) .
Revised: ( x ) for every f ∈ F and therefore that (( i j ) · x, x ) ∈ ρ ( F ) .

Original: By assumption, we infer that (( i j ) · x, x ) ∈
Revised: By assumption, we infer that (( i j ) · x, x ) ∈

Original: ρ ( h ) too, i.e. that h (( i j ) · x ) = h ( x ) and so that h ( x ) i
Revised: ρ ( h ) too, i.e. that h (( i j ) · x ) = h ( x ) and so that h ( x ) i

Original: = h ( x ) j by equivariance, which concludes our proof.
Revised: = h ( x ) j by equivariance, which concludes our proof.

Original: D. R EDUCTIONS FOR GNN S
Revised: D. R EDUCTIONS FOR GNN S

Original: We now present a lemma which explains how to instantiate the two corollaries above in the case of
Revised: We now present a lemma which explains how to instantiate the two corollaries above in the case of

Original: GNNs by replacing continuous functions with MLPs.
Revised: GNNs by replacing continuous functions with MLPs.

Original: Lemma 32.
Revised: Lemma 32.

Original: Fix X
Revised: Fix X

Original: some compact space, n ≥ 1 and F a ﬁnite-dimensional feature space.
Revised: some compact space, n ≥ 1 and F a ﬁnite-dimensional feature space.

Original: Let
Revised: Let

Original: F 0 ⊆ (cid:83) ∞
Revised: F 0 ⊆ (cid:83) ∞

Original: h =1 C ( X, R h ) be stable by concatenation and consider,
Revised: h =1 C ( X, R h ) be stable by concatenation and consider,

Original: [Equation]
Revised: [Equation]

Original: Then, if E ( F ) ⊆ C ( X, F ) is the set of functions obtained by replacing the MLP m in the deﬁnition of
Revised: Then, if E ( F ) ⊆ C ( X, F ) is the set of functions obtained by replacing the MLP m in the deﬁnition of

Original: F by an arbitrary continuous function, E ( F ) satisﬁes,
Revised: F by an arbitrary continuous function, E ( F ) satisﬁes,

Original: F = E ( F ) 2.
Revised: F = E ( F ) 2.

Original: ρ ( F ) =
Revised: ρ ( F ) =

Original: ρ ( E ( F )) 3.
Revised: ρ ( E ( F )) 3.

Original: For any h ∈ C ( F 2 , F ) , f, g ∈ E ( F ) , x (cid:55)→ ( h ( f ( x ) 1 , g ( x ) 1 ) , . . .
Revised: For any h ∈ C ( F 2 , F ) , f, g ∈ E ( F ) , x (cid:55)→ ( h ( f ( x ) 1 , g ( x ) 1 ) , . . .

Original: , h ( f ( x ) n , g ( x ) n ))
Revised: , h ( f ( x ) n , g ( x ) n ))

Original: ∈ E ( F ) .
Revised: ∈ E ( F ) .

Original: If, for any f ∈ F , x (cid:55)→
Revised: If, for any f ∈ F , x (cid:55)→

Original: (cid:32) n (cid:88) i =
Revised: (cid:32) n (cid:88) i =

Original: f ( x ) i , n (cid:88) i
Revised: f ( x ) i , n (cid:88) i

Original: = f
Revised: = f

Original: ( x ) i . .
Revised: ( x ) i . .

Original: . , n (cid:88)
Revised: . , n (cid:88)

Original: i =
Revised: i =

Original: f ( x ) i (cid:33) ∈ F , then, for any f ∈ E ( F ) , x (cid:55)→ (cid:32) n (cid:88)
Revised: f ( x ) i (cid:33) ∈ F , then, for any f ∈ E ( F ) , x (cid:55)→ (cid:32) n (cid:88)

Original: i =
Revised: i =

Original: f ( x ) i , n (cid:88) i
Revised: f ( x ) i , n (cid:88) i

Original: = f
Revised: = f

Original: ( x ) i . .
Revised: ( x ) i . .

Original: . , n (cid:88)
Revised: . , n (cid:88)

Original: i =
Revised: i =

Original: f ( x ) i (cid:33)
Revised: f ( x ) i (cid:33)

Original: ∈ E ( F ) .
Revised: ∈ E ( F ) .

Original: If F is equivariant w.r.t.
Revised: If F is equivariant w.r.t.

Original: the action described in Cor. 30, so is E ( F ) .
Revised: the action described in Cor. 30, so is E ( F ) .

Original: Proof.
Revised: Proof.

Original: Deﬁne,
Revised: Deﬁne,

Original: [Equation]
Revised: [Equation]

Original: As MLP are continuous and by the universality of MLP on a compact set (see Section C.1),
Revised: As MLP are continuous and by the universality of MLP on a compact set (see Section C.1),

Original: [Equation]
Revised: [Equation]

Original: F .
Revised: F .

Original: This already implies 1.
Revised: This already implies 1.

Original: and that ρ (cid:0) F (cid:1) ⊆
Revised: and that ρ (cid:0) F (cid:1) ⊆

Original: ρ ( E ( F )) ⊆
Revised: ρ ( E ( F )) ⊆

Original: ρ ( F ) .
Revised: ρ ( F ) .

Original: Using Lem.
Revised: Using Lem.

Original: 15 yields 2.
Revised: 15 yields 2.

Original: We now show 3.
Revised: We now show 3.

Original: Take h ∈ C ( F 2 , F ) , f, g ∈ F 0 , f ∈ C
Revised: Take h ∈ C ( F 2 , F ) , f, g ∈ F 0 , f ∈ C

Original: ( X, R h f ) ,
Revised: ( X, R h f ) ,

Original: g ∈ C
Revised: g ∈ C

Original: ( X, R h g ) and m ∈ C
Revised: ( X, R h g ) and m ∈ C

Original: ( R h f , F ) , l ∈ C
Revised: ( R h f , F ) , l ∈ C

Original: ( R h g , F ) .
Revised: ( R h g , F ) .

Original: All we have to show is that,
Revised: All we have to show is that,

Original: [Equation]
Revised: [Equation]

Original: But as F 0 is stable by concatenation, x (cid:55)→ ( f ( x ) , g ( x )
Revised: But as F 0 is stable by concatenation, x (cid:55)→ ( f ( x ) , g ( x )

Original: ) ∈
Revised: ) ∈

Original: R h
Revised: R h

Original: f
Revised: f

Original: + h g is still in F 0 .
Revised: + h g is still in F 0 .

Original: Moreover, y ∈ R h
Revised: Moreover, y ∈ R h

Original: f + h g
Revised: f + h g

Original: (cid:55)→ h
Revised: (cid:55)→ h

Original: ( m ( y 1 , . . .
Revised: ( m ( y 1 , . . .

Original: y h f ) , l ( y h f +1 , . . .
Revised: y h f ) , l ( y h f +1 , . . .

Original: , y h
Revised: , y h

Original: f
Revised: f

Original: + h g ) is also in C
Revised: + h g ) is also in C

Original: ( R h f
Revised: ( R h f

Original: + h g , F ) which shows that the map above is indeed in E ( F ) .
Revised: + h g , F ) which shows that the map above is indeed in E ( F ) .

Original: The last two points are immediate consequences of the deﬁnition of E ( F ) .
Revised: The last two points are immediate consequences of the deﬁnition of E ( F ) .

Original: Thm. 5 and Thm. 6 are now obtained by combining Lem.
Revised: Thm. 5 and Thm. 6 are now obtained by combining Lem.

Original: 32 with Cor.
Revised: 32 with Cor.

Original: 29 and Cor.
Revised: 29 and Cor.

Original: E
Revised: E

Original: [Equation]
Revised: [Equation]

Original: FOR EXPRESSIVENESS OF GNN S
Revised: FOR EXPRESSIVENESS OF GNN S

Original: Note that in the Theorem 6, the additional stability assumption is “almost” necessary to obtain the result.
Revised: Note that in the Theorem 6, the additional stability assumption is “almost” necessary to obtain the result.

Original: Indeed, if the result holds, i.e.,
Revised: Indeed, if the result holds, i.e.,

Original: [Equation]
Revised: [Equation]

Original: then one can show, that, if f ∈ F , then
Revised: then one can show, that, if f ∈ F , then

Original: [Equation]
Revised: [Equation]

Original: Indeed, f ∈ F so that it has a weaker discriminating power than F , so that ρ ( F ) =
Revised: Indeed, f ∈ F so that it has a weaker discriminating power than F , so that ρ ( F ) =

Original: ρ ( F 0 ) ⊆
Revised: ρ ( F 0 ) ⊆

Original: ρ ( f ) .
Revised: ρ ( f ) .

Original: But, by construction of ˜ f , ρ ( f ) ⊆
Revised: But, by construction of ˜ f , ρ ( f ) ⊆

Original: ρ (cid:16) ˜ f (cid:17) so that ρ ( F 0 ) ⊆
Revised: ρ (cid:16) ˜ f (cid:17) so that ρ ( F 0 ) ⊆

Original: ρ (cid:16) ˜ f (cid:17) .
Revised: ρ (cid:16) ˜ f (cid:17) .

Original: As ˜ f is also in C E ( X, F n ) , ˜ f ∈ F .
Revised: As ˜ f is also in C E ( X, F n ) , ˜ f ∈ F .

Original: E. E XPRESSIVITY OF GNN LAYERS
Revised: E. E XPRESSIVITY OF GNN LAYERS

Original: Lemma 33.
Revised: Lemma 33.

Original: Fix F 0 , F 1 (non-trivial) ﬁnite dimensionnal vector spaces.
Revised: Fix F 0 , F 1 (non-trivial) ﬁnite dimensionnal vector spaces.

Original: Consider the action of
Revised: Consider the action of

Original: G = S n on F 0 × F n × kdeﬁned by,
Revised: G = S n on F 0 × F n × kdeﬁned by,

Original: [Equation]
Revised: [Equation]

Original: Let K ⊆ F 0 × F n × kbe a compact set.
Revised: Let K ⊆ F 0 × F n × kbe a compact set.

Original: Then, the set of functions from K ⊆ F 0 × F n × kto F 1 of the form, (
Revised: Then, the set of functions from K ⊆ F 0 × F n × kto F 1 of the form, (

Original: [Equation]
Revised: [Equation]

Original: (cid:88)
Revised: (cid:88)

Original: j = k (cid:89)
Revised: j = k (cid:89)

Original: w
Revised: w

Original: [Equation]
Revised: [Equation]

Original: 
Revised: 

Original:  where f 0 : F 0 × R h → F 1 , f j : F 0 → R h , j = 1 , . .
Revised:  where f 0 : F 0 × R h → F 1 , f j : F 0 → R h , j = 1 , . .

Original: , k are multi-linear perceptrons and h ≥ 1 , is dense in C
Revised: , k are multi-linear perceptrons and h ≥ 1 , is dense in C

Original: I ( K, F 1 ) .
Revised: I ( K, F 1 ) .

Original: Proof.
Revised: Proof.

Original: Denote by F ⊆ C
Revised: Denote by F ⊆ C

Original: I ( K, F 1 ) the set of such functions.
Revised: I ( K, F 1 ) the set of such functions.

Original: To prove that F = C I ( K, F 1 ) , we ﬁrst apply Thm. 5.
Revised: To prove that F = C I ( K, F 1 ) , we ﬁrst apply Thm. 5.

Original: We get, that,
Revised: We get, that,

Original: [Equation]
Revised: [Equation]

Original: We now characterize ρ ( F ) .
Revised: We now characterize ρ ( F ) .

Original: Actually, it is equal to ρ ( inv ) =
Revised: Actually, it is equal to ρ ( inv ) =

Original: { (( x 0 , x ) , ( y 0 , y )) ∈ (cid:0) F 0 × F n × k(cid:1) 2 : ∃ σ ∈ S n , ( x 0 , x ) = σ · ( y 0 , y ) } .
Revised: { (( x 0 , x ) , ( y 0 , y )) ∈ (cid:0) F 0 × F n × k(cid:1) 2 : ∃ σ ∈ S n , ( x 0 , x ) = σ · ( y 0 , y ) } .

Original: As the functions of F are invariant
Revised: As the functions of F are invariant

Original: , ρ ( inv ) ⊆
Revised: , ρ ( inv ) ⊆

Original: ρ ( F ) .
Revised: ρ ( F ) .

Original: We now show the reverse.
Revised: We now show the reverse.

Original: Take ( x 0 , x ) , ( y 0 , y ) ∈ F 0 × F n × ksuch that there does not exist σ ∈ S n such that ( x 0 , x ) = σ · ( y 0 , y ) .
Revised: Take ( x 0 , x ) , ( y 0 , y ) ∈ F 0 × F n × ksuch that there does not exist σ ∈ S n such that ( x 0 , x ) = σ · ( y 0 , y ) .

Original: If x 0 (cid:54) =
Revised: If x 0 (cid:54) =

Original: y 0 , there exists f 0 : F 0 → F 1 MLP such that f ( x 0 ) (cid:54) =
Revised: y 0 , there exists f 0 : F 0 → F 1 MLP such that f ( x 0 ) (cid:54) =

Original: f ( y 0 )
Revised: f ( y 0 )

Original: so that (( x 0 , x ) , ( y 0 , y )) / ∈ ρ ( F ) .
Revised: so that (( x 0 , x ) , ( y 0 , y )) / ∈ ρ ( F ) .

Original: Otherwise, there does not exists σ ∈ S n such that ( x σ − 1 (1) , . . .
Revised: Otherwise, there does not exists σ ∈ S n such that ( x σ − 1 (1) , . . .

Original: , x σ − 1 ( n ) ) = ( y 1 , . .
Revised: , x σ − 1 ( n ) ) = ( y 1 , . .

Original: y n ) by deﬁnition of the action of G = S n .
Revised: y n ) by deﬁnition of the action of G = S n .

Original: Now, apply Lem.
Revised: Now, apply Lem.

Original: 31 with F ← F k 0 , the universality of MLP and use the decomposition given to get f j : F 0 → R h , j = 1 , . . .
Revised: 31 with F ← F k 0 , the universality of MLP and use the decomposition given to get f j : F 0 → R h , j = 1 , . . .

Original: k such that
Revised: k such that

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: kw =1 f w ( y j,w ) .
Revised: kw =1 f w ( y j,w ) .

Original: Choosing an appropriate MLP f 0 : R h → Fyields that (( x 0 , x ) , ( y 0 , y )) / ∈ ρ ( F ) .
Revised: Choosing an appropriate MLP f 0 : R h → Fyields that (( x 0 , x ) , ( y 0 , y )) / ∈ ρ ( F ) .

Original: Hence, we have shown that,
Revised: Hence, we have shown that,

Original: [Equation]
Revised: [Equation]

Original: E. A PPROXIMATION THEOREMS FOR GNN S
Revised: E. A PPROXIMATION THEOREMS FOR GNN S

Original: We now have all the tools to ﬁnally prove our main result
Revised: We now have all the tools to ﬁnally prove our main result

Original: [Equation]
Revised: [Equation]

Original: Theorem 34.
Revised: Theorem 34.

Original: Let K discr ⊆
Revised: Let K discr ⊆

Original: G n × F n 0 , K ⊆ F n 2 0 be compact sets.
Revised: G n × F n 0 , K ⊆ F n 2 0 be compact sets.

Original: For the invariant case, we have:
Revised: For the invariant case, we have:

Original: [Equation]
Revised: [Equation]

Original: For the equivariant case
Revised: For the equivariant case

Original: [Equation]
Revised: [Equation]

Original: we have:
Revised: we have:

Original: [Equation]
Revised: [Equation]

Original: We decompose the proof with an additional lemma
Revised: We decompose the proof with an additional lemma

Original: [Equation]
Revised: [Equation]

Original: Lemma 35.
Revised: Lemma 35.

Original: Let K discr ⊆
Revised: Let K discr ⊆

Original: G n × F n 0 , K ⊆ F n 2 0 be compact sets.
Revised: G n × F n 0 , K ⊆ F n 2 0 be compact sets.

Original: For the invariant case
Revised: For the invariant case

Original: [Equation]
Revised: [Equation]

Original: and any k ≥ 2 , we have,
Revised: and any k ≥ 2 , we have,

Original: [Equation]
Revised: [Equation]

Original: For the equivariant case, and any k ≥ 2 , we have:
Revised: For the equivariant case, and any k ≥ 2 , we have:

Original: [Equation]
Revised: [Equation]

Original: Proof of Thm.
Revised: Proof of Thm.

Original: The theorem is now a direct consequence of Prop. 13 and Lem.
Revised: The theorem is now a direct consequence of Prop. 13 and Lem.

Original: We now move to the proof of Lem.
Revised: We now move to the proof of Lem.

Original: Proof of Lem.
Revised: Proof of Lem.

Original: First, focus on the invariant case.
Revised: First, focus on the invariant case.

Original: Let F denote MGNN I , k
Revised: Let F denote MGNN I , k

Original: -LGNN
Revised: -LGNN

Original: I or k
Revised: I or k

Original: -FGNN I and X be either K discr or K so that X is compact and F
Revised: -FGNN I and X be either K discr or K so that X is compact and F

Original: ⊆ C
Revised: ⊆ C

Original: I ( X, F ) .
Revised: I ( X, F ) .

Original: Applying Thm.
Revised: Applying Thm.

Original: 5 directly gives,
Revised: 5 directly gives,

Original: [Equation]
Revised: [Equation]

Original: { f ∈ C
Revised: { f ∈ C

Original: I
Revised: I

Original: [Equation]
Revised: [Equation]

Original: X,
Revised: X,

Original: [Equation]
Revised: [Equation]

Original: which is the desired result.
Revised: which is the desired result.

Original: We now move to the equivariant case.
Revised: We now move to the equivariant case.

Original: First, let us replace MGNN E by another class, which is slightly simpler to analyze.
Revised: First, let us replace MGNN E by another class, which is slightly simpler to analyze.

Original: Deﬁne,
Revised: Deﬁne,

Original: [Equation]
Revised: [Equation]

Original: It holds that ρ ( MGNN E (cid:48) )
Revised: It holds that ρ ( MGNN E (cid:48) )

Original: =
Revised: =

Original: ρ ( MGNN E ) and,
Revised: ρ ( MGNN E ) and,

Original: [Equation]
Revised: [Equation]

Original: Therefore, if we show that,
Revised: Therefore, if we show that,

Original: [Equation]
Revised: [Equation]

Original: we will have the desired result.
Revised: we will have the desired result.

Original: Let F denote MGNN E (cid:48) , k -LGNN E or k -FGNN E and X be either K discr or K so that X is compact and F
Revised: Let F denote MGNN E (cid:48) , k -LGNN E or k -FGNN E and X be either K discr or K so that X is compact and F

Original: ⊆ C E ( X, F n ) .
Revised: ⊆ C E ( X, F n ) .

Original: We now wish to apply Thm.
Revised: We now wish to apply Thm.

Original: [Equation]
Revised: [Equation]

Original: but we need to verify the stability assumption ﬁrst.
Revised: but we need to verify the stability assumption ﬁrst.

Original: Thus, we show that, for any f ∈ F ,
Revised: Thus, we show that, for any f ∈ F ,

Original: [Equation]
Revised: [Equation]

Original: • If F = MGNN E (cid:48) .Take
Revised: • If F = MGNN E (cid:48) .Take

Original: f ∈ MGNN E (cid:48) .
Revised: f ∈ MGNN E (cid:48) .

Original: f is of the form,
Revised: f is of the form,

Original: [Equation]
Revised: [Equation]

Original: 1 , where F t : F nt → F nt +1 are message passing layers, F T +1 = F and λ ∈ { 0 , 1 } .
Revised: 1 , where F t : F nt → F nt +1 are message passing layers, F T +1 = F and λ ∈ { 0 , 1 } .

Original: We need to show that there is a MGNN E (cid:48) which implements,
Revised: We need to show that there is a MGNN E (cid:48) which implements,

Original: [Equation]
Revised: [Equation]

Original: If λ = 1 , f is exactly the function of (26).
Revised: If λ = 1 , f is exactly the function of (26).

Original: Otherwise, if λ = 0 , we build another MGNN E (cid:48) implementing this function.
Revised: Otherwise, if λ = 0 , we build another MGNN E (cid:48) implementing this function.

Original: Denote by F T +1 : F n → F n the (simple) message passing layer deﬁned by F T +1 ( h ) i = m E ( h i ) for i ∈ [ n ] and any h ∈ F n .
Revised: Denote by F T +1 : F n → F n the (simple) message passing layer deﬁned by F T +1 ( h ) i = m E ( h i ) for i ∈ [ n ] and any h ∈ F n .

Original: Then, the MGNN E (cid:48) (1 − λ (cid:48) )
Revised: Then, the MGNN E (cid:48) (1 − λ (cid:48) )

Original: Id + λ (cid:48) S 1 ) ◦ F T +1 ◦ F T ◦ . . .
Revised: Id + λ (cid:48) S 1 ) ◦ F T +1 ◦ F T ◦ . . .

Original: F 2 ◦ F 1 with λ (cid:48)
Revised: F 2 ◦ F 1 with λ (cid:48)

Original: =
Revised: =

Original: 1 exactly implements (26).
Revised: 1 exactly implements (26).

Original: If F = k -LGNN
Revised: If F = k -LGNN

Original: E
Revised: E

Original: A function f of
Revised: A function f of

Original: this
Revised: this

Original: class
Revised: class

Original: is
Revised: is

Original: of the form,
Revised: of the form,

Original: [Equation]
Revised: [Equation]

Original: where F t : F n k t → F n k t +1 are linear graph layers and F T
Revised: where F t : F n k t → F n k t +1 are linear graph layers and F T

Original: +1 = F .
Revised: +1 = F .

Original: Our goal is to show that there is a GNN of k -LGNN E which implements,
Revised: Our goal is to show that there is a GNN of k -LGNN E which implements,

Original: [Equation]
Revised: [Equation]

Original: By deﬁnition of linear graph layers, the map F T +1 : F n k → F n k deﬁned by, for G ∈ F n k , ∀
Revised: By deﬁnition of linear graph layers, the map F T +1 : F n k → F n k deﬁned by, for G ∈ F n k , ∀

Original: ( i 1 , . .
Revised: ( i 1 , . .

Original: , i k ) ∈ [ n ] k ,
Revised: , i k ) ∈ [ n ] k ,

Original: F T +1 ( G ) i 1 ,...,i k
Revised: F T +1 ( G ) i 1 ,...,i k

Original: = m E ( S k 1 ( G ) i ) , is a linear graph layer as deﬁned in Section 3.
Revised: = m E ( S k 1 ( G ) i ) , is a linear graph layer as deﬁned in Section 3.

Original: Now, consider, the linear graph layer,
Revised: Now, consider, the linear graph layer,

Original: F T +2 : F n k → F n k deﬁned by, for G ∈ F n k , ∀
Revised: F T +2 : F n k → F n k deﬁned by, for G ∈ F n k , ∀

Original: ( i 1 , . .
Revised: ( i 1 , . .

Original: , i k ) ∈ [ n ] k ,
Revised: , i k ) ∈ [ n ] k ,

Original: F T +1 ( G ) i 1 ,...,i k
Revised: F T +1 ( G ) i 1 ,...,i k

Original: = n (cid:88) i =
Revised: = n (cid:88) i =

Original: G i,i 2 ,...,i k .
Revised: G i,i 2 ,...,i k .

Original: Then, the k -LGNN F T +2 ◦ F T +1 ◦ F T ◦ . . .
Revised: Then, the k -LGNN F T +2 ◦ F T +1 ◦ F T ◦ . . .

Original: F 2 ◦ F 1 ◦ I k exactly implements (27).
Revised: F 2 ◦ F 1 ◦ I k exactly implements (27).

Original: • If F = k -FGNN E .
Revised: • If F = k -FGNN E .

Original: A function f of this class is of the form,
Revised: A function f of this class is of the form,

Original: [Equation]
Revised: [Equation]

Original: where F t : F n k t → F n k t +1 are FGL (see Section 3) and F T +1 = F .
Revised: where F t : F n k t → F n k t +1 are FGL (see Section 3) and F T +1 = F .

Original: We build a GNN of k -MGNN E which implements,
Revised: We build a GNN of k -MGNN E which implements,

Original: [Equation]
Revised: [Equation]

Original: For w ∈ [ k ] , deﬁne the FGL H w : F n k → F n k by,
Revised: For w ∈ [ k ] , deﬁne the FGL H w : F n k → F n k by,

Original: [Equation]
Revised: [Equation]

Original: Then H 2 ◦ · · · ◦ H k : F n k → F n k computes the sum of the elements of the input tensor over the last k − 1 dimensions like to S k 1 : F n k → F n and H 1 ◦ H 2 ◦ · · · ◦ H k : F n k → F n k computes the full sum of the elements of the input tensor like to S k : F n k → F .
Revised: Then H 2 ◦ · · · ◦ H k : F n k → F n k computes the sum of the elements of the input tensor over the last k − 1 dimensions like to S k 1 : F n k → F n and H 1 ◦ H 2 ◦ · · · ◦ H k : F n k → F n k computes the full sum of the elements of the input tensor like to S k : F n k → F .

Original: Finally, consider the FGL F T +1 : F n k → F n k associated to m E , i.e. such that, for any G ∈ F n k , ∀ ( i 1 , . .
Revised: Finally, consider the FGL F T +1 : F n k → F n k associated to m E , i.e. such that, for any G ∈ F n k , ∀ ( i 1 , . .

Original: , i k ) ∈ [ n ] k ,
Revised: , i k ) ∈ [ n ] k ,

Original: F T +1 ( G ) i 1 ,...,i k
Revised: F T +1 ( G ) i 1 ,...,i k

Original: = m E
Revised: = m E

Original: ( G i 1 ,...,i k ) .
Revised: ( G i 1 ,...,i k ) .

Original: Now, the k -FGNN
Revised: Now, the k -FGNN

Original: [Equation]
Revised: [Equation]

Original: H k ◦ F T +1 ◦ F T ◦ . . .
Revised: H k ◦ F T +1 ◦ F T ◦ . . .

Original: F 2 ◦ F 1 ◦
Revised: F 2 ◦ F 1 ◦

Original: I k exactly implements (28).
Revised: I k exactly implements (28).

Original: F E XTENSION TO GRAPHS OF VARYING SIZES F. E XTENSION TO DISCONNECTED INPUT SPACES
Revised: F E XTENSION TO GRAPHS OF VARYING SIZES F. E XTENSION TO DISCONNECTED INPUT SPACES

Original: Here we show that our results can be extended to graphs of varying sizes similarly to Keriven &
Revised: Here we show that our results can be extended to graphs of varying sizes similarly to Keriven &

Original: Peyré (2019).
Revised: Peyré (2019).

Original: There are two ways to do it.
Revised: There are two ways to do it.

Original: The ﬁrst would be to directly adapt all the proofs but it would make them more cumbersome.
Revised: The ﬁrst would be to directly adapt all the proofs but it would make them more cumbersome.

Original: Instead, we extend them with a simple argument presented below.
Revised: Instead, we extend them with a simple argument presented below.

Original: As a side beneﬁt, this general lemma makes it possible to extend almost all the approximation results for graph neural networks from the literature.
Revised: As a side beneﬁt, this general lemma makes it possible to extend almost all the approximation results for graph neural networks from the literature.

Original: The abstract setting is the following.
Revised: The abstract setting is the following.

Original: Given a compact input space X , assume that there is some ﬁnite set A , and X α , α
Revised: Given a compact input space X , assume that there is some ﬁnite set A , and X α , α

Original: ∈
Revised: ∈

Original: A a family of pairwise disjoints compact sets such that X
Revised: A a family of pairwise disjoints compact sets such that X

Original: =
Revised: =

Original: (cid:70)
Revised: (cid:70)

Original: α ∈
Revised: α ∈

Original: A X α .
Revised: A X α .

Original: Crucially, we will assume that the X α are in distinct connected components.
Revised: Crucially, we will assume that the X α are in distinct connected components.

Original: Intuitively, they do not "touch" each other.
Revised: Intuitively, they do not "touch" each other.

Original: Similarly, assume that the output space Y can be written as Y = (cid:70) α ∈
Revised: Similarly, assume that the output space Y can be written as Y = (cid:70) α ∈

Original: A Y α , with
Revised: A Y α , with

Original: Y α , α ∈
Revised: Y α , α ∈

Original: A a family of (pairwise disjoints) real vector spaces.
Revised: A a family of (pairwise disjoints) real vector spaces.

Original: Before moving to the results, we need a last deﬁnition.
Revised: Before moving to the results, we need a last deﬁnition.

Original: Informally, we need that the functions f that we will consider do not change the number of nodes of their inputs.
Revised: Informally, we need that the functions f that we will consider do not change the number of nodes of their inputs.

Original: Formally, we say that f : X −→ Y is adapted if f ( X α ) ⊆
Revised: Formally, we say that f : X −→ Y is adapted if f ( X α ) ⊆

Original: Y α for each α ∈ A , and denote by C ad ( X, Y )
Revised: Y α for each α ∈ A , and denote by C ad ( X, Y )

Original: the of continuous and adapted functions from X to Y .
Revised: the of continuous and adapted functions from X to Y .

Original: We can now state our lemma to adapt our results to graphs with varying node sizes.
Revised: We can now state our lemma to adapt our results to graphs with varying node sizes.

Original: Lemma 36.
Revised: Lemma 36.

Original: Let X be a compact space, Y be a topological space and A a ﬁnite set.
Revised: Let X be a compact space, Y be a topological space and A a ﬁnite set.

Original: Assume that there exists X α , α ∈
Revised: Assume that there exists X α , α ∈

Original: A a family of pairwise disjoints compact sets such that X
Revised: A a family of pairwise disjoints compact sets such that X

Original: =
Revised: =

Original: (cid:70)
Revised: (cid:70)

Original: α ∈
Revised: α ∈

Original: A X α and the
Revised: A X α and the

Original: X α ’s are in distinct connected components.
Revised: X α ’s are in distinct connected components.

Original: Also assume that there is Y α ,
Revised: Also assume that there is Y α ,

Original: α ∈
Revised: α ∈

Original: A a family of (pairwise disjoints) real vector spaces such that Y =
Revised: A a family of (pairwise disjoints) real vector spaces such that Y =

Original: (cid:70)
Revised: (cid:70)

Original: α ∈
Revised: α ∈

Original: A Y α .
Revised: A Y α .

Original: Consider F ⊆ C ad ( X, Y ) a (non-empty) set of adapted functions and, for each α ∈ A , deﬁne F | X α = { f | X α : f ∈ F} ⊆ C
Revised: Consider F ⊆ C ad ( X, Y ) a (non-empty) set of adapted functions and, for each α ∈ A , deﬁne F | X α = { f | X α : f ∈ F} ⊆ C

Original: ( X α , Y α ) the restriction of the functions of F to X α .
Revised: ( X α , Y α ) the restriction of the functions of F to X α .

Original: Assume that the following holds,
Revised: Assume that the following holds,

Original: Each F | X α is a sub-algebra of C ( X α , Y α ) which contains the constant function 1 Y α . 2.
Revised: Each F | X α is a sub-algebra of C ( X α , Y α ) which contains the constant function 1 Y α . 2.

Original: There is a set of functions F scal ⊆
Revised: There is a set of functions F scal ⊆

Original: C ( X, R ) such that F scal .
Revised: C ( X, R ) such that F scal .

Original: F ⊆ F and it discriminates between the X α ’s, i.e. ρ ( F scal ) ⊆
Revised: F ⊆ F and it discriminates between the X α ’s, i.e. ρ ( F scal ) ⊆

Original: (cid:70)
Revised: (cid:70)

Original: α ∈
Revised: α ∈

Original: A X 2 α .
Revised: A X 2 α .

Original: Then the closure of F is,
Revised: Then the closure of F is,

Original: [Equation]
Revised: [Equation]

Original: for the distance on C ad ( X, Y ) deﬁned by, for f, g ∈ C ad ( X, Y ) ,
Revised: for the distance on C ad ( X, Y ) deﬁned by, for f, g ∈ C ad ( X, Y ) ,

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: Deﬁne the set of functions S ⊆ C ( X, R ) by,
Revised: Deﬁne the set of functions S ⊆ C ( X, R ) by,

Original: [Equation]
Revised: [Equation]

Original: By deﬁnition, F scal ⊆ S .
Revised: By deﬁnition, F scal ⊆ S .

Original: Moreover, as each F | X α is a subalgebra, S is also a subalgebra of C ( X, R ) .
Revised: Moreover, as each F | X α is a subalgebra, S is also a subalgebra of C ( X, R ) .

Original: As X is compact, we can apply Cor.
Revised: As X is compact, we can apply Cor.

Original: 17 to S to get that, in C ( X, R ) ,
Revised: 17 to S to get that, in C ( X, R ) ,

Original: [Equation]
Revised: [Equation]

Original: In the ﬁrst part of the proof, we check that, for each α , the function f α : X −→ [0 , 1] deﬁned by
Revised: In the ﬁrst part of the proof, we check that, for each α , the function f α : X −→ [0 , 1] deﬁned by

Original: [Equation]
Revised: [Equation]

Original: α | X
Revised: α | X

Original: [Equation]
Revised: [Equation]

Original: is continuous and satisfy ρ ( S ) ⊆
Revised: is continuous and satisfy ρ ( S ) ⊆

Original: ρ ( f α ) .
Revised: ρ ( f α ) .

Original: This will means that such functions belong to F scal .
Revised: This will means that such functions belong to F scal .

Original: As the X β ’s are in different connected components, it is enough to check its continuity on each X β .
Revised: As the X β ’s are in different connected components, it is enough to check its continuity on each X β .

Original: Indeed, each f α | X β is constant so continuous.
Revised: Indeed, each f α | X β is constant so continuous.

Original: The second fact comes from the second assumption.
Revised: The second fact comes from the second assumption.

Original: Indeed, take ( x, y ) ∈
Revised: Indeed, take ( x, y ) ∈

Original: X such that ( x, y ) ∈
Revised: X such that ( x, y ) ∈

Original: ρ ( S ) .
Revised: ρ ( S ) .

Original: As F scal ⊆ S , in particular ( x, y ) ∈
Revised: As F scal ⊆ S , in particular ( x, y ) ∈

Original: ρ ( F scal ) .
Revised: ρ ( F scal ) .

Original: But, by assumption, ρ ( F scal ) ⊆ (cid:70) β ∈
Revised: But, by assumption, ρ ( F scal ) ⊆ (cid:70) β ∈

Original: A X 2 β .
Revised: A X 2 β .

Original: Thus, necessarily x and y belong to the same X β so that f α ( x ) =
Revised: Thus, necessarily x and y belong to the same X β so that f α ( x ) =

Original: f α ( y ) .
Revised: f α ( y ) .

Original: Therefore, we conclude that f α ∈ S .
Revised: Therefore, we conclude that f α ∈ S .

Original: We now prove the announced equality.
Revised: We now prove the announced equality.

Original: By the deﬁnition of the distance, the ﬁrst inclusion, F ⊆ (cid:8) f ∈ C ad ( X, Y ) :
Revised: By the deﬁnition of the distance, the ﬁrst inclusion, F ⊆ (cid:8) f ∈ C ad ( X, Y ) :

Original: f | X α ∈ F α (cid:9) is immediate.
Revised: f | X α ∈ F α (cid:9) is immediate.

Original: We focus on the other way.
Revised: We focus on the other way.

Original: Take h ∈ C ad ( X, Y ) such that h | X α ∈ F α for every α ∈ A and (cid:15) > 0 .
Revised: Take h ∈ C ad ( X, Y ) such that h | X α ∈ F α for every α ∈ A and (cid:15) > 0 .

Original: By deﬁnition of h , there exists, for each α ∈ A , g α ∈ F such that,
Revised: By deﬁnition of h , there exists, for each α ∈ A , g α ∈ F such that,

Original: [Equation]
Revised: [Equation]

Original: As the X α ’s are compact and the g α ’s are continuous, max
Revised: As the X α ’s are compact and the g α ’s are continuous, max

Original: α ∈
Revised: α ∈

Original: A sup X α (cid:107) g
Revised: A sup X α (cid:107) g

Original: α (cid:107) <
Revised: α (cid:107) <

Original: + ∞ and denote by
Revised: + ∞ and denote by

Original: M > 0 a bound on this quantity.
Revised: M > 0 a bound on this quantity.

Original: We have shown above that each f α is in S so there exists, for each α ∈ A , l α ∈ S such that,
Revised: We have shown above that each f α is in S so there exists, for each α ∈ A , l α ∈ S such that,

Original: [Equation]
Revised: [Equation]

Original: By deﬁnition of S and, as F is a subalgebra, (cid:80) α ∈
Revised: By deﬁnition of S and, as F is a subalgebra, (cid:80) α ∈

Original: A l α g α ∈ F and, for each β ∈ A ,
Revised: A l α g α ∈ F and, for each β ∈ A ,

Original: [Equation]
Revised: [Equation]

Original: which concludes the proof.
Revised: which concludes the proof.

Original: F. A PPROXIMATION THEOREM WITH VARYING GRAPH SIZE
Revised: F. A PPROXIMATION THEOREM WITH VARYING GRAPH SIZE

Original: We now state our theorem in the case of varying graph size like Keriven & Peyré (2019).
Revised: We now state our theorem in the case of varying graph size like Keriven & Peyré (2019).

Original: With
Revised: With

Original: Lem.
Revised: Lem.

Original: 36, it is indeed straightforward to extend any approximation result initially proven for a class of graphs of ﬁxed size.
Revised: 36, it is indeed straightforward to extend any approximation result initially proven for a class of graphs of ﬁxed size.

Original: However, as the complete proof would require new notations again, we only give a sketch of proof.
Revised: However, as the complete proof would require new notations again, we only give a sketch of proof.

Original: Fix N ≥ 1 and consider the space of graphs (described by tensors) of size less than N , F ≤ N 2= (cid:83)
Revised: Fix N ≥ 1 and consider the space of graphs (described by tensors) of size less than N , F ≤ N 2= (cid:83)

Original: Nn =1 F n 2 0 .
Revised: Nn =1 F n 2 0 .

Original: Equip this space with the ﬁnal topology or, equivalently, the graph edit distance.
Revised: Equip this space with the ﬁnal topology or, equivalently, the graph edit distance.

Original: Then, the last thing to check to apply Lem.
Revised: Then, the last thing to check to apply Lem.

Original: 36 is that the classes of GNN that we consider indeed discriminate between graphs of different sizes, which is immediate.
Revised: 36 is that the classes of GNN that we consider indeed discriminate between graphs of different sizes, which is immediate.

Original: Likewise, for message passing GNN, consider G ≤ N × F ≤ N
Revised: Likewise, for message passing GNN, consider G ≤ N × F ≤ N

Original: [Equation]
Revised: [Equation]

Original: , with a similar topology or the graph edit distance
Revised: , with a similar topology or the graph edit distance

Original: [Equation]
Revised: [Equation]

Original: Corollary 37.
Revised: Corollary 37.

Original: Let K discr ⊆
Revised: Let K discr ⊆

Original: G ≤ N × F ≤ N 0 , K ⊆ F ≤ N 2be compact sets.
Revised: G ≤ N × F ≤ N 0 , K ⊆ F ≤ N 2be compact sets.

Original: For the invariant case, we have:
Revised: For the invariant case, we have:

Original: [Equation]
Revised: [Equation]

Original: For the equivariant case
Revised: For the equivariant case

Original: [Equation]
Revised: [Equation]

Original: we have:
Revised: we have:

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: This corollary is a direct consequence of Thm.
Revised: This corollary is a direct consequence of Thm.

Original: 34 and Lem.
Revised: 34 and Lem.

Original: 36, using Lem.
Revised: 36, using Lem.

Original: 32 to satisfy the sub-algebra assumption of Lem.
Revised: 32 to satisfy the sub-algebra assumption of Lem.