Reduce use of UIP as suggested by Robbert

theories/algebra/ofe.v

@@ -1287,40 +1287,43 @@ Arguments sigO {_} _.
 
 (** Ofe for [sigT]. The first component must be discrete
     and use Leibniz equality, while the second component might be any OFE. *)
-Section sigmaT.
+Section sigT.
   Import EqNotations.
-  Local Set Default Proof Using "Type*".
-  (* For this 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]). *)
 
-  Context {A : Type} `{!∀ a b : A, ProofIrrel (a = b)} {P : A → ofeT}.
+  Context {A : Type} {P : A → ofeT}.
   Implicit Types x : sigT P.
 
   (**
-    The equality/distance for [{ a : A & P }] uses Leibniz equality on [A] to
+    The distance for [{ a : A & P }] uses Leibniz equality on [A] to
     transport the second components to the same type,
-    and then step-indexed equality/distance on the second component.
+    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_equiv : Equiv (sigT P) := λ x1 x2,
-    ∃ eq : projT1 x1 = projT1 x2, rew eq in projT2 x1 ≡ projT2 x2.
   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 https://github.com/coq/coq/issues/3814. *)
-  Definition sigT_equiv_eq x1 x2 : (x1 ≡ x2) ↔
-    ∃ eq : projT1 x1 = projT1 x2, rew eq in projT2 x1 ≡ projT2 x2 :=
+  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 :=
+    ∃ eq : projT1 x1 = projT1 x2, (rew eq in projT2 x1) ≡{n}≡ projT2 x2 :=
       reflexivity _.
 
-  Definition sigT_equiv_proj1 x y : x ≡ y → projT1 x = projT1 y := proj1_ex.
   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} :
@@ -1328,17 +1331,15 @@ Section sigmaT.
       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).
   Proof.
-    split.
-    - move => [x Px] [y Py] /=.
-      setoid_rewrite sigT_dist_eq; rewrite sigT_equiv_eq /=; split.
-      + destruct 1 as [-> Heq]. exists eq_refl => /=.
-        by apply equiv_dist.
-      + intros Heq. destruct (Heq 0) as [-> _]. exists eq_refl => /=.
-        apply equiv_dist => n. case: (Heq n) => [Hrefl].
-        have -> //=: Hrefl = eq_refl y. exact: proof_irrel.
-    - move => n; split; hnf; setoid_rewrite sigT_dist_eq.
+    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.
@@ -1346,7 +1347,7 @@ Section sigmaT.
         destruct 1 as [-> Heq1].
         destruct 1 as [-> Heq2]. exists eq_refl => /=. by trans y.
     - setoid_rewrite sigT_dist_eq.
-      move => n [xa x] [ya y] /=. destruct 1 as [-> Heq].
+      move => [xa x] [ya y] /=. destruct 1 as [-> Heq].
       exists eq_refl. exact: dist_S.
   Qed.
 
@@ -1356,9 +1357,8 @@ Section sigmaT.
 
   Global Instance sigT_discrete x : Discrete (projT2 x) → Discrete x.
   Proof.
-    move: x => [xa x] ? [ya y].
-    rewrite sigT_dist_eq sigT_equiv_eq /=. destruct 1 as [-> Hxy].
-    exists eq_refl. move: Hxy => /=. exact: discrete _.
+    move: x => [xa x] ? [ya y] [] /=; intros -> => /= Hxy n.
+    exists eq_refl => /=. apply equiv_dist, (discrete _), Hxy.
   Qed.
 
   Global Instance sigT_ofe_discrete : (∀ a, OfeDiscrete (P a)) → OfeDiscrete sigTO.
@@ -1367,6 +1367,24 @@ Section sigmaT.
   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.
+  Proof.
+    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.
+  Qed.
+
+  (* 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.
@@ -1378,13 +1396,10 @@ Section sigmaT.
       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; cbn.
-    have {Heqi0} -> /= : Heqi0 = Heqin; first exact: proof_irrel.
-    exact Hgoal.
+      destruct Heqn0.
+      by rewrite /= (proof_irrel Heqi0 Heqin).
     Qed.
 
-  Section cofe.
-    Context `{!∀ a, Cofe (P a)}.
     Definition sigT_compl : Compl sigTO :=
       λ c, existT (projT1 (chain_car c 0)) (compl (chain_map_snd c)).
 
@@ -1398,12 +1413,12 @@ Section sigmaT.
       destruct (sigT_chain_const_proj1 c n); simpl. done.
     Qed.
   End cofe.
-End sigmaT.
+End sigT.
 
-Arguments sigTO {_ _} _.
+Arguments sigTO {_} _.
 
 Section sigTOF.
-  Context {A : Type} `{!∀ x y : A, ProofIrrel (x = y)} .
+  Context {A : Type}.
 
   Program Definition sigT_map {P1 P2 : A → ofeT} :
     discrete_funO (λ a, P1 a -n> P2 a) -n>
@@ -1425,10 +1440,10 @@ Section sigTOF.
     repeat intro. exists eq_refl => /=. solve_proper.
   Qed.
   Next Obligation.
-    intros; exists eq_refl => /=. apply oFunctor_id.
+    simpl; intros. apply (existT_proper eq_refl), oFunctor_id.
   Qed.
   Next Obligation.
-    intros; exists eq_refl => /=. apply oFunctor_compose.
+    simpl; intros. apply (existT_proper eq_refl), oFunctor_compose.
   Qed.
 
   Global Instance sigTOF_contractive {F} :
@@ -1437,7 +1452,7 @@ Section sigTOF.
     repeat intro. apply sigT_map => a. exact: oFunctor_contractive.
   Qed.
 End sigTOF.
-Arguments sigTOF {_ _} _%OF.
+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.