Get rid of `Peano.nat.eq_nat_dec` in favour of `decide`.

`decide` works much better because of the general lemmas and tacitcs

theories/F_mu_ref_conc/lang.v

@@ -107,7 +107,7 @@ Module lang.
     match op,v1,v2 with
     | Add, LitV (Nat a), LitV (Nat b) => Some $ LitV (Nat (a + b))
     | Sub, LitV (Nat a), LitV (Nat b) => Some $ LitV (Nat (a - b))
-    | Eq, LitV (Nat a), LitV (Nat b) => Some $ if eq_nat_dec a b then LitV (Bool true) else LitV (Bool false)
+    | Eq, LitV (Nat a), LitV (Nat b) => Some $ if decide (a = b) then LitV (Bool true) else LitV (Bool false)
     | Eq, LitV (Bool a), LitV (Bool b) => Some $ LitV (Bool (eqb a b))
     | Eq, LitV (Loc l1), LitV (Loc l2) => Some $ if decide (l1 = l2) then LitV (Bool true) else LitV (Bool false)
     | Le, LitV (Nat a), LitV (Nat b) => Some $ if le_dec a b then LitV (Bool true) else LitV (Bool false)

theories/examples/generative.v

@@ -133,13 +133,13 @@ Section namegen_refinement.
       iDestruct (bij_agree with "HB Hln Hl'n'") as %Hag.
       destruct (decide (l = l')) as [->|Hll].
       + assert (n = n') as -> by (apply Hag; auto).
-        destruct (PeanoNat.Nat.eq_dec n' n'); last congruence.
+        case_decide; last by contradiction.
         iMod ("Hcl" with "[-]") as "_".
         { iNext. iExists _,_; iFrame. }
         iApply bin_log_related_bool.
       + assert (n ≠ n') as Hnn'.
         { intros Hnn. apply Hll. by apply Hag. }
-        destruct (PeanoNat.Nat.eq_dec n n'); first congruence.
+        case_decide; first by contradiction.
         iMod ("Hcl" with "[-]") as "_".
         { iNext. iExists _,_; iFrame. }
         iApply bin_log_related_bool.

theories/examples/ticket_lock.v

@@ -227,12 +227,10 @@ Section refinement.
         iDestruct "Hlk" as (o n' b) "(>Hlo & >Hln & Hseq & Hl' & Hrest)".
         iModIntro. iExists _; iFrame; iNext.
         iIntros "Hlo".
-        destruct (decide (n = o)); simplify_eq; rel_op_l.
+        rel_op_l.
+        case_decide; subst; rel_if_l.
         (* The ticket is called out *)
-        * replace (if PeanoNat.Nat.eq_dec o o then LitV true else LitV false) with #true; last first.
-          { (* TODO :( *) case_match; congruence. }
-          rel_if_l.
-          unlock lock.acquire.
+        * unlock lock.acquire.
           rel_rec_r.
           destruct b.
           { iDestruct (own_valid_2 with "Hticket Hrest") as %?%gset_disj_valid_op.
@@ -243,10 +241,7 @@ Section refinement.
             iApply "Hpool". iExists _,_,_; by iFrame. }
           iApply bin_log_related_unit.
 
-        * replace (if PeanoNat.Nat.eq_dec n o then LitV true else LitV false) with #false; last first.
-          { (* TODO :( *) case_match; congruence. }
-          rel_if_l.
-          iMod ("Hcl" with "[-Hticket]") as "_".
+        * iMod ("Hcl" with "[-Hticket]") as "_".
           { iNext. iExists P; iFrame.
             iApply "Hpool". iExists _,_,_; by iFrame. }
           rel_rec_l.

theories/logrel/fundamental_binary.v

@@ -282,8 +282,7 @@ Section masked.
     rel_op_l; eauto.
     rel_op_r; eauto.
     value_case.
-    destruct op; simpl in Hopv'; simplify_eq/=; try destruct eq_nat_dec; try destruct le_dec;
-      try destruct lt_dec; eauto.
+    destruct op; simpl in Hopv'; simplify_eq/=; try case_match; eauto.
   Qed.
 
   Lemma bin_log_related_bool_binop Δ Γ op e1 e2 e1' e2' τ :