Split off liftn_sig, add lift{3,4}_sig

2 files changed, 51 insertions, 0 deletions

diff --git a/src/Util/Sigma/Associativity.v b/src/Util/Sigma/Associativity.v
new file mode 100644
index 000000000..fbebf2584
--- /dev/null
+++ b/src/Util/Sigma/Associativity.v
@@ -0,0 +1,18 @@
+(** * Reassociation of [sig] *)
+Definition sig_sig_assoc {A} {P : A -> Prop} {Q}
+  : { a : A | P a /\ Q a } -> { ap : { a : A | P a } | Q (proj1_sig ap) }
+  := fun apq => exist _ (exist _ (proj1_sig apq) (proj1 (proj2_sig apq))) (proj2 (proj2_sig apq)).
+Ltac sig_sig_assoc :=
+  lazymatch goal with
+  | [ |- { a : ?A | ?P } ]
+    => let P'' := fresh a in
+       let P' := fresh P'' in
+       let term := constr:(fun a : A => match P with
+                                        | P' => ltac:(let v := (eval cbv [P'] in P') in
+                                                      lazymatch eval pattern (proj1_sig a) in v with
+                                                      | ?P _ => exact P
+                                                      end)
+                                        end) in
+       let Q := lazymatch (eval cbv beta in term) with fun _ => ?term => term end in
+       apply (@sig_sig_assoc _ _ Q)
+  end.
diff --git a/src/Util/Sigma/Lift.v b/src/Util/Sigma/Lift.v
new file mode 100644
index 000000000..c57172029
--- /dev/null
+++ b/src/Util/Sigma/Lift.v
@@ -0,0 +1,33 @@
+(** * Lift foralls out of sig proofs *)
+
+Definition lift1_sig {A C} (P:A->C->Prop)
+           (op_sig : forall (a:A), {c | P a c})
+: { op : A -> C | forall (a:A), P a (op a) }
+:= exist
+     (fun op => forall a, P a (op a))
+     (fun a => proj1_sig (op_sig a))
+     (fun a => proj2_sig (op_sig a)).
+
+Definition lift2_sig {A B C} (P:A->B->C->Prop)
+           (op_sig : forall (a:A) (b:B), {c | P a b c})
+  : { op : A -> B -> C | forall (a:A) (b:B), P a b (op a b) }
+  := exist
+       (fun op => forall a b, P a b (op a b))
+       (fun a b => proj1_sig (op_sig a b))
+       (fun a b => proj2_sig (op_sig a b)).
+
+Definition lift3_sig {A B C D} (P:A->B->C->D->Prop)
+           (op_sig : forall (a:A) (b:B) (c:C), {d | P a b c d})
+  : { op : A -> B -> C -> D | forall (a:A) (b:B) (c:C), P a b c (op a b c) }
+  := exist
+       (fun op => forall a b c, P a b c (op a b c))
+       (fun a b c => proj1_sig (op_sig a b c))
+       (fun a b c => proj2_sig (op_sig a b c)).
+
+Definition lift4_sig {A B C D E} (P:A->B->C->D->E->Prop)
+           (op_sig : forall (a:A) (b:B) (c:C) (d:D), {e | P a b c d e})
+  : { op : A -> B -> C -> D -> E | forall (a:A) (b:B) (c:C) (d:D), P a b c d (op a b c d) }
+  := exist
+       (fun op => forall a b c d, P a b c d (op a b c d))
+       (fun a b c d => proj1_sig (op_sig a b c d))
+       (fun a b c d => proj2_sig (op_sig a b c d)).