Commit 43fb8444 authored by Ralf Jung's avatar Ralf Jung
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Merge branch 'sigT-cFunctor' into 'master'

Cofe and cFunctor for sigT

See merge request iris/iris!278
parents 472aa3bc d043dd23
......@@ -208,6 +208,8 @@ s/\bgnameC/gnameO/g;
' $(find theories -name "*.v")
- Add a COFE construction (and functor) on dependent pairs `sigTO`, dual to
## Iris 3.1.0 (released 2017-12-19)
......@@ -1284,3 +1284,175 @@ Section sigma.
End sigma.
Arguments sigO {_} _.
(** Ofe for [sigT]. The first component must be discrete
and use Leibniz equality, while the second component might be any OFE. *)
Section sigT.
Import EqNotations.
Context {A : Type} {P : A ofeT}.
Implicit Types x : sigT P.
The distance for [{ a : A & P }] uses Leibniz equality on [A] to
transport the second components to the same type,
and then step-indexed distance on the second component.
Unlike in the topos of trees, with (C)OFEs we cannot use step-indexed equality
on the first component.
Instance sigT_dist : Dist (sigT P) := λ n x1 x2,
eq : projT1 x1 = projT1 x2, rew eq in projT2 x1 {n} projT2 x2.
Usually we'd give a direct definition, and show it equivalent to
[∀ n, x1 ≡{n}≡ x2] when proving the [equiv_dist] OFE axiom.
But here the equivalence requires UIP — see [sigT_equiv_eq_alt].
By defining [equiv] in terms of [dist], we can define an OFE
without assuming UIP, at the cost of complex reasoning on [equiv].
Instance sigT_equiv : Equiv (sigT P) := λ x1 x2,
n, x1 {n} x2.
(** Unfolding lemmas.
Written with [↔] not [=] to avoid *)
Definition sigT_equiv_eq x1 x2 : (x1 x2) n, x1 {n} x2 :=
reflexivity _.
Definition sigT_dist_eq x1 x2 n : (x1 {n} x2)
eq : projT1 x1 = projT1 x2, (rew eq in projT2 x1) {n} projT2 x2 :=
reflexivity _.
Definition sigT_dist_proj1 n {x y} : x {n} y projT1 x = projT1 y := proj1_ex.
Definition sigT_equiv_proj1 x y : x y projT1 x = projT1 y := λ H, proj1_ex (H 0).
(** [existT] is "non-expansive". *)
Lemma existT_ne n {i1 i2} {v1 : P i1} {v2 : P i2} :
(eq : i1 = i2), (rew f_equal P eq in v1 {n} v2)
existT i1 v1 {n} existT i2 v2.
Proof. intros ->; simpl. exists eq_refl => /=. done. Qed.
Lemma existT_proper {i1 i2} {v1 : P i1} {v2 : P i2} :
(eq : i1 = i2), (rew f_equal P eq in v1 v2)
existT i1 v1 existT i2 v2.
Proof. intros eq Heq n. apply (existT_ne n eq), equiv_dist, Heq. Qed.
Definition sigT_ofe_mixin : OfeMixin (sigT P).
split => // n.
- split; hnf; setoid_rewrite sigT_dist_eq.
+ intros. by exists eq_refl.
+ move => [xa x] [ya y] /=. destruct 1 as [-> Heq].
by exists eq_refl.
+ move => [xa x] [ya y] [za z] /=.
destruct 1 as [-> Heq1].
destruct 1 as [-> Heq2]. exists eq_refl => /=. by trans y.
- setoid_rewrite sigT_dist_eq.
move => [xa x] [ya y] /=. destruct 1 as [-> Heq].
exists eq_refl. exact: dist_S.
Canonical Structure sigTO : ofeT := OfeT (sigT P) sigT_ofe_mixin.
Implicit Types (c : chain sigTO).
Global Instance sigT_discrete x : Discrete (projT2 x) Discrete x.
move: x => [xa x] ? [ya y] [] /=; intros -> => /= Hxy n.
exists eq_refl => /=. apply equiv_dist, (discrete _), Hxy.
Global Instance sigT_ofe_discrete : ( a, OfeDiscrete (P a)) OfeDiscrete sigTO.
Proof. intros ??. apply _. Qed.
Lemma sigT_chain_const_proj1 c n : projT1 (c n) = projT1 (c 0).
Proof. refine (sigT_dist_proj1 _ (chain_cauchy c 0 n _)). lia. Qed.
Lemma sigT_equiv_eq_alt `{! a b : A, ProofIrrel (a = b)} x1 x2 :
x1 x2
eq : projT1 x1 = projT1 x2, rew eq in projT2 x1 projT2 x2.
setoid_rewrite equiv_dist; setoid_rewrite sigT_dist_eq; split => Heq.
- move: (Heq 0) => [H0eq1 _].
exists H0eq1 => n. move: (Heq n) => [] Hneq1.
by rewrite (proof_irrel H0eq1 Hneq1).
- move: Heq => [Heq1 Heqn2] n. by exists Heq1.
(* For this COFE construction we need UIP (Uniqueness of Identity Proofs)
on [A] (i.e. [∀ x y : A, ProofIrrel (x = y)]. UIP is most commonly obtained
from decidable equality (by Hedberg’s theorem, see
[stdpp.proof_irrel.eq_pi]). *)
Section cofe.
Context `{! a b : A, ProofIrrel (a = b)} `{! a, Cofe (P a)}.
Program Definition chain_map_snd c : chain (P (projT1 (c 0))) :=
{| chain_car n := rew (sigT_chain_const_proj1 c n) in projT2 (c n) |}.
Next Obligation.
move => c n i Hle /=.
(* [Hgoal] is our thesis, up to casts: *)
case: (chain_cauchy c n i Hle) => [Heqin Hgoal] /=.
(* Pretty delicate. We have two casts to [projT1 (c 0)].
We replace those by one cast. *)
move: (sigT_chain_const_proj1 c i) (sigT_chain_const_proj1 c n)
=> Heqi0 Heqn0.
(* Rewrite [projT1 (c 0)] to [projT1 (c n)] in goal and [Heqi0]: *)
destruct Heqn0.
by rewrite /= (proof_irrel Heqi0 Heqin).
Definition sigT_compl : Compl sigTO :=
λ c, existT (projT1 (chain_car c 0)) (compl (chain_map_snd c)).
Global Program Instance sigT_cofe : Cofe sigTO := { compl := sigT_compl }.
Next Obligation.
intros n c. rewrite /sigT_compl sigT_dist_eq /=.
exists (symmetry (sigT_chain_const_proj1 c n)).
(* Our thesis, up to casts: *)
pose proof (conv_compl n (chain_map_snd c)) as Hgoal.
move: (compl (chain_map_snd c)) Hgoal => pc0 /=.
destruct (sigT_chain_const_proj1 c n); simpl. done.
End cofe.
End sigT.
Arguments sigTO {_} _.
Section sigTOF.
Context {A : Type}.
Program Definition sigT_map {P1 P2 : A ofeT} :
discrete_funO (λ a, P1 a -n> P2 a) -n>
sigTO P1 -n> sigTO P2 :=
λne f xpx, existT _ (f _ (projT2 xpx)).
Next Obligation.
move => ?? f n [x px] [y py] [/= Heq]. destruct Heq; simpl.
exists eq_refl => /=. by f_equiv.
Next Obligation.
move => ?? n f g Heq [x px] /=. exists eq_refl => /=. apply Heq.
Program Definition sigTOF (F : A oFunctor) : oFunctor := {|
oFunctor_car A CA B CB := sigTO (λ a, oFunctor_car (F a) A _ B CB);
oFunctor_map A1 _ A2 _ B1 _ B2 _ fg := sigT_map (λ a, oFunctor_map (F a) fg)
Next Obligation.
repeat intro. exists eq_refl => /=. solve_proper.
Next Obligation.
simpl; intros. apply (existT_proper eq_refl), oFunctor_id.
Next Obligation.
simpl; intros. apply (existT_proper eq_refl), oFunctor_compose.
Global Instance sigTOF_contractive {F} :
( a, oFunctorContractive (F a)) oFunctorContractive (sigTOF F).
repeat intro. apply sigT_map => a. exact: oFunctor_contractive.
End sigTOF.
Arguments sigTOF {_} _%OF.
Notation "{ x & P }" := (sigTOF (λ x, P%OF)) : oFunctor_scope.
Notation "{ x : A & P }" := (@sigTOF A%type (λ x, P%OF)) : oFunctor_scope.
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