Blame and Coercion: Together Again for the First Time

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1 Blame and Coercion: Together Again for the First Time Supplementary Material Jeremy Siek Indiana University, USA Peter Thiemann Universität Freiburg, Germany Philip Wadler University of Edinburgh, UK Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. PLDI 15, June 13 17, 2015, Portland, OR, USA. Copyright c 2015 ACM /15/06... $

2 A. Positive and negative subtyping Lemma 1 (Positive and negative subtyping). 1. A <: + B iff A p = B BC safe C p. 2. A <: B iff A p = B BC safe C p. Proof. A <: + B implies A = p B BC safe C p and A <: B implies A = p B BC safe C p is proved by mutual induction on the definition of A = p B BC. s for positive subtyping: p ι = ι BC = id ι satisfies ι <: + ι and id ι safe C p. A B = p A B BC = A p = A BC B = p B BC. From the assumption A B <: + A B, we obtain A <: A and B <: + B. By induction, we get that A p = A BC safe C p and B = p B BC safe C p, which proves the claim. = p BC = id satisfies <: + and id safe C p. G = p BC = G!. Immediate because G <: +. A = p BC = A = p G BC ; G! where A, A G, and A G. Hence, it must be that G = and A = A B so that A = p G BC = A B p = BC = = p A BC B p = BC. Since <: A and B <: +, the result holds by induction. = p G BC. Not applicable because : + G. = p A BC where A, A G, and A G. Not applicable because : + A. s for negative subtyping: p ι = ι BC = id ι satisfies ι <: ι and id ι safe C p. A B = p A B BC = A p = A BC B = p B BC. From the assumption A B <: A B, we obtain A <: + A and B <: B. By induction, we get that A p = A BC safe C p and B = p B BC safe C p, which proves the claim. = p BC = id satisfies <: and id safe C p. G = p BC = G!. Immediate because G <:. A = p BC = A = p G BC ; G!. If A <:, then it must be that A <: G. Hence, the claim holds by induction. = p G BC = G? p is safe for p and <: G holds. = p B BC = G? p ; G = p B BC (where B, B G, and G B). <: B is satisfied regardless of B. Hence, it must be that G = so that B = A B and we need to examine = p A B BC = A p = BC = p B BC. As A <: + and <: B we can argue by induction that A p = BC safe C p and = p B BC safe C p. The reverse implication is proved by similar mutual induction on the definition of the translation.

3 B. Bisimulation between coercions and threesomes Here we give the full proof of Proposition 16. Lemma 2 (Compose Identity Threesomes). s id A CS = s and id A CS s = s Proof. The proof is a straightforward induction on s and A. Lemma 3. If M s and t V 2, then M s t V 2. Proof of Proposition 16. Part 1 and 2. We proceed by case analysis on M M, in each case proving the two statements: 1. If M C N then M S N and N N for some N. 2. If M S N then M C N and N N for some N. (Here we assume parts 3 and 4, which we later prove independently.) Both statements are vacuously true because k cannot reduce. k k M M op( M) op( M ) 1. op( k) op( M ) op( k) δ(op, k) δ(op, k) 2. op( M) op( k) op( k) δ(op, k) δ(op, k) x x Both statements are vacuously true because x cannot reduce. M M λx:a. M λx:a. M Both statements are vacouously true because lambda terms cannot reduce. M 1 M 1 M 2 M 2 M 1 M 2 M 1 M 2 1. We proceed by case analysis on M = M 1 M 2 C N. So either M 1 reduces, M 2 reduces, or they are both values. Suppose M 1 reduces, i.e., M 1 C M 3. From M 1 M 2 M, we have M = M 1 M 2 and M 1 M 1 and M 2 M 2. By the induction hypothesis, M 1 S M 3 and M 3 M 3. So M 1 M 2 S M 3 M 2 and M 3 M 2 M 3 M 2. The case for M 2 reducing is essentialy the same as for M 1 reducing. Suppose M 1 and M 2 are values. We proceed by cases on M 1. M 1 = k: M cannot reduce; M 1 = λx:a. M 11: part of beta redex, see (a) below; M 1 = V c d : part of coercion redex, see (b) below; M 1 = V G! : M cannot reduce. Let V 2 = M 2. (a) (λx:a. M 11) V 2 C M 11[x := V ] We have (λx:a. M 11) V 2 M 1 M 2 then proceed by case analysis on (λx:a. M 11) V 1. V 2

4 Subcase M 11 M 11 λx:a. M 11 λx:a. M 11 (λx:a. M 11) V 2 (λx:a. M 11) V 2 Subcase λx:a. M 11 U λx:a. M 11 U id A B CS M 11[x := V 2] M 11[x := V 2] (λx:a. M 11) V 2 U id A B CS V 2 (U V 2 id A CS ) id B CS ((λx:a. M 11) V 2 id A CS ) id B CS M 11[x := V 2] M 11[x := V 2 id A CS ] id B CS (b) (V c d ) W C (V W c ) d We proceed by induction on V c d M 1. There are two cases to consider. (Rule (iii) does not apply because the premise would relate a value to a function application.) Subcase rule (i). V c d M 11 V c d : A B V c d M 11 id A id B We have M 11 1 and V c d 1 by induction. So we have 1 = U (s 1 s 2) c d CS and V U s 1 s 2. (V c d ) W The left is related to the right by rule (iii). Subcase rule (ii). because (M 11 id A id B ) M 2 U (s 1 s 2) c d CS id A id B V 2 U (s 1 s 2) c d CS (id A id B) V 2 Lemma 2 (V W c ) d (U (s 1 s 2) c d CS ) V (V c d ) W (V W c ) d V M 11 s (M 11 s c CS d CS ) M 2 W M 2 W c M 2 c CS V W c (M 11 s ) (M 2 c CS ) (V W c ) d (M 11 s c CS d CS ) M 2 2. We proceed by case analysis on M 1 M 2 S N. (a) (λx:a. M 11) V 2 S M 11[x := V 2] M 1 M 2 (λx:a. M 11) V 2 (λx:a. M 11) V M 11[x := V 2] M 11[x := V 2] 2 2

5 (b) (U s t ) W S (U W s ) t M 1 M 2 ( c n d n c 1 d 1 ) V 2 (U s t ) W ( (V 2 c 1 c n )) d n d 1 (U W s ) t M 1 M 1 c CS = s M 1 c M 1 s 1. We proceed by case analysis on M 1 c C N. (a) id A C V1 ida M 1 id A CS (b) G! G? p C V1 G! G? p V 1 id A CS M 1 id G CS ; G! G? p ; id G CS (c) G! H? p C blame p V 1 id G CS ; G! G? p ; id G CS V G! H? p M 1 id G CS ; G! H? p ; id H CS 1 M 1 p A B blame p blame p (d) c; d C c d (e) p A B C blame p 2. We proceed by case analysis on M 1 t S N. c; d c d p A B M 1 t M 1 p A B blame p blame p (a) U id ι S U M1 idι id ι U id ι U

6 (b) U id S U M 1 id id U id (c) M 2 s t S M 2 s t U M 1 c M 2 s t We have M 1 M 2 s and therefore M 1 c M 2 s t. (d) U p A B S blame p M 1 c M 2 s t M 1 p A B p A B U p A B blame p blame p M 1 M 1 s c CS = t M 1 c M 1 s t 1. We proceed by case analysis on M 1 c C N. (a) id A C id A (b) G! G? p C G! G? p M 1 s id A CS M 1 s G! G? p (c) G! H? p C blame p G! H? p M 1 s M 1 s G! H? p ; id H CS M 1 p A B (d) c; d C c d (e) p A B C blame p blame p blame p c; d c d p A B M 1 s t M 1 s p A B M 1 p A B blame p blame p

7 2. We proceed by case analysis on M 1 s t S N. (a) U id ι S U. There are two cases for s t = id ι: i. s = t = id ι M 1 id ι id ι U id ι U ii. s = id ι; ι! and t = ι? p ; id ι. In that case, the assumption is M 1 U id ι; ι!. By inversion, M 1 = M 11 ι! and M 11 U id ι. By further inversion, M 11 U. Hence: M 11 ι! ι? p ι! ι? p U id ι U (b) U id S U M 1 id id U id (c) M 2 s s t S M 2 s s t U M 1 c M 2 s s t M 1 c M 2 s s t We have M 1 M 2 s s and therefore M 1 M 2 s s. With t CS = c we conclude M 1 c M 2 s s t. (d) U p A B S blame p There are three ways that we could have s t = p A B. i. s = (g; G!), t = (H? p ; i) M 1 c U p A B G! H? p blame p blame p ii. s = p A B We have M 1 U p A B so by the induction hypothesis M1 blame p. M 1 c U p A B blame p blame p iii. t = p A B M 1 p A B p A B blame p U p A B blame p

8 M 1 M 1 s M 2 t 1 d CS = t 2 M 1 d M 1 s (t 1 t 2) M 2 1. We proceed by case analysis on M 1 d C N, but every case is vacuously true because they require M 1 to be a value, but M 1 corresponds to a function application. 2. We proceed by cases on M 1 s (t 1 t 2) M 2 S N. We have M 1 U s 1 s 2 W t 1. So M 1 = (M 2 M 3 c c 1 ) d 1 where c d CS = t 1 t 2 and c 1 d 1; ; c n d n CS = s 1 s 2. (M 2.. c k d k M 3 c c 1..).. d 1 d (V 2.. c k d k V 3 c c 1..).. d 1 d U s 1 s 2 t 1 t 2 W (V 2 W c c 1.. c n ) d n.. d 1 d (U W t 1 s 1 ) s 2 t 2 Part 3. We show that the term M on the right can become a value V that corresponds to V. We proceed by induction on V. V = k. We proceed by cases on k M, but we only have one case to consider. Subcase k k Take V = k. V = λx:a. N. We proceed by induction on (λx:a. N) M. Subcase N N λx:a. N λx:a. N We take V = λx:a. N. λx:a. N M 1 id A B CS = id A id B Subcase λx:a. N M 1 id A id B By the inner induction hypothesis we have the following. Now suppose = λx:a. N. Then id A id B is a value. On the other hand, suppose = U s t. λx:a. N M 1 id A id B V 1 id A id B λx:a. N U s t id A id B U s t V = G!. We proceed by induction on G! M. There is one case to consider. (Rule (iii) does not apply because the premises would relate a value to a function application.) Subcase rule (i) G! M 1 G! M 1 id CS G! M 1 id V 1 id Subcase rule (ii) G! V M 1 s G! M 1 s G! CS 1

9 The inner induction hypothesis gives us V1 M 1 s Suppose = k. Then k G! CS is a value. By Lemma 3 we have k G! M 1 s G! CS k G! CS Suppose V 1 = λx:a. N. Then (λx:a. N ) G! CS is a value. By Lemma 3 we have G! M 1 s G! CS (λx:a. N ) G! CS Suppose = U g; H!. Then has type, but that contradicts it having type G. Suppose = U s t. We have G! U s t G! CS By Lemma 3 we conclude U (s t ); G! G! M 1 s G! CS U (s t ); G! V = c d. We proceed by induction on c d M. There are three cases to consider. (Rule (iii) does not apply because the premise would relate a value to a function application.) Subcase rule (i) c d M 1 c d : A B id A B CS = t c d M 1 t We have M 1 and c d by the inner induction hypothesis. We proceed by cases on with the knowledge that it is of function type. Suppose = λx:a. e. Then id A id B is a value and we relate the left to the right by rule (i). Suppose = U c d. c d U c d id A CS id A CS Subcase rule (ii) M 1 s c d CS = t U (c d ) ( id A CS id A CS ) Lemma 2 c d U c d c d M 1 s t We have M 1 s and by the inner induction hypothesis. Then applying some case analysis on we have c CS d CS V and c d V for some V. Part 4. We show that the term M on the left can become a value that corresponds to V. We proceed by induction on V. V = k. By inversion on M k we have M = k, which is already a value, so we take V = M. V = λx:a. N. By inversion on M λx:a. N we have M = λx:a. N and take V = M. V = U s t. Inversion of M U s t gives us two cases to consider. Subcase for rule (i) M U M : A id A CS = s t M U s t

10 By the induction hypothesis, M C V where V U. Then the left and right sides are related by rule (i). Subcase for rule (ii). M 1 U s c CS = t M 1 c U s t We have M = M 1 c and (s t ) = s t. By the induction hypothesis, M 1 C where U s. We proceed with a nested induction on c. Suppose c = id A. id A U s id A CS Lemma 2 U s Suppose c = G!. Then t = G! CS = id G CS ; G!, but that contradicts (s t ) = s t. Suppose c = G? p. Then t = G? p ; id G CS. With (s t ) = s t, we have s = (s t); G!. Then from U (s t); G! we have = V 2 G! with V 2 U s t for some V 2. So we obtain: G? p V 2 G! G? p U (s t); G! G? p ; id G CS Lemma 2 V 2 U s t Next suppose c = c 1 c 2, then c 1 c 2 is already a value. From U s and c CS = t we have c U s t by rule (ii). Suppose c = (c 1; c 2). We have t = c 1 CS c 2 CS. We obtain the following with two uses of the the inner induction hypothesis. c 1; c 2 U s t c 1 c 2 IH V 2 c 2 IH V 3 U s c 1 CS c 2 CS Suppose c = p A B. Then t = p A B and (s t ) = p B, but (s t ) = s t so we have a contradiction. V = U g; G!. Considering M U g; G!, only rule (ii) applies. Subcase (ii): IH IH M 1 U s c CS = t M 1 c U s t By the induction hypothesis, we have M 1 C and U s. We proceed by nested induction on c. Suppose c = id. id U s id CS U s Lemma 2 Suppose c = H!. Then we have H! U s H! CS. Suppose c = H? p. Then t = H? p CS = H? p ; id H CS. But that contradicts (s t) = (g; G!). Suppose c = c 1 c 2. Then t = c 1 c 2 CS = c 1 CS c 2 CS. But that contradicts (s t) = (g; G!).

11 Suppose c = (c 1; c 2). We use the same reasoning as for the corresponding case in V = U s t, that is, we obtain the following with two uses of the the inner induction hypothesis. c 1; c 2 c 1 c 2 IH V 2 c 2 IH V 3 U s t U s c 1 CS c 2 CS Suppose c = p A B. Then t = p A B and (s t ) = p B, but (s t ) = (g; G!) so we have a contradiction. Part 5 and 6. blame p blame p IH IH

12 C. Translation is bisimilar Here we sketch the proof of Proposition 17. Proposition 17. M M CS. Proof. (Sketch). By induction on M. The only non-trivial case is for M c where we need to apply rules (i) and (ii) to establish. In all other cases, the congruence rules are sufficient.