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Require Import Utf8.
Unset Elimination Schemes.
Definition Decision (P : Prop) := {P} + {¬ P}.
Definition RelDecision {A B : Type} (R : A → B → Prop) :=
∀ (x : A) (y : B), Decision (R x y) : Type.
Inductive gmap_dep_ne (A : Type) : Type :=
GNode001 : gmap_dep_ne A → gmap_dep_ne A
| GNode010 : A → gmap_dep_ne A
| GNode011 : A → gmap_dep_ne A → gmap_dep_ne A
| GNode100 : gmap_dep_ne A → gmap_dep_ne A
| GNode101 : gmap_dep_ne A → gmap_dep_ne A → gmap_dep_ne A
| GNode110 : gmap_dep_ne A → A → gmap_dep_ne A
| GNode111 : gmap_dep_ne A → A → gmap_dep_ne A → gmap_dep_ne A.
Variant gmap_dep (A : Type) : Type :=
GEmpty : gmap_dep A | GNodes : gmap_dep_ne A → gmap_dep A.
Arguments GEmpty {A}.
Arguments GNodes {A}.
Record gmap {K : Type} (EqDecision0 : RelDecision (@eq K))
(A : Type) : Type := GMap { gmap_car : gmap_dep A }.
Inductive gtest {K : Type} (H : RelDecision (@eq K)) := GTest : gmap H (gtest H) → gtest H.
Arguments GTest {_ _} _.
Definition option_union_with (A : Type) (f : A → A → option A) (mx my : option A) :=
match mx with
| Some x => match my with
| Some y => f x y
| None => Some x
end
| None => match my with
| Some y => Some y
| None => None
end
end.
Definition gmap_dep_ne_case (A : Type) (B : Type) (t : gmap_dep_ne A)
(f : gmap_dep A → option A → gmap_dep A → B) :=
match t with
| GNode001 _ r => f GEmpty None (GNodes r)
| GNode010 _ x => f GEmpty (Some (x)) GEmpty
| GNode011 _ x r => f GEmpty (Some (x)) (GNodes r)
| GNode100 _ l => f (GNodes l) None GEmpty
| GNode101 _ l r => f (GNodes l) None (GNodes r)
| GNode110 _ l x => f (GNodes l) (Some (x)) GEmpty
| GNode111 _ l x r => f (GNodes l) (Some (x)) (GNodes r)
end.
Definition gmap_dep_omap_aux {A B : Type} (go : gmap_dep_ne A → gmap_dep B)
(tm : gmap_dep A) :=
match tm with
| GEmpty => GEmpty
| GNodes t' => go t'
end.
Definition option_bind {A B : Type} (f : A → option B) (mx : option A) :=
match mx with
| Some x => f x
| None => None
end.
Definition GNode (A : Type)
(ml : gmap_dep A)
(mx : option (A))
(mr : gmap_dep A) :=
match ml with
| GEmpty =>
match mx with
| Some x =>
match mr with
| GEmpty => GNodes (GNode010 _ x)
| GNodes r => GNodes (GNode011 _ x r)
end
| None =>
match mr with
| GEmpty => GEmpty
| GNodes r => GNodes (GNode001 _ r)
end
end
| GNodes l =>
match mx with
| Some (x) =>
match mr with
| GEmpty => GNodes (GNode110 _ l x)
| GNodes r => GNodes (GNode111 _ l x r)
end
| None =>
match mr with
| GEmpty => GNodes (GNode100 _ l)
| GNodes r => GNodes (GNode101 _ l r)
end
end
end.
Definition gmap_dep_ne_omap (A B : Type) (f : A → option B) :=
fix go (t : gmap_dep_ne A) {struct t} : gmap_dep B :=
gmap_dep_ne_case _ _ t
(λ (ml : gmap_dep A) (mx : option A) (mr : gmap_dep A),
GNode _
(gmap_dep_omap_aux (go) ml)
(option_bind f mx)
(gmap_dep_omap_aux (go) mr)).
Definition gmap_merge_aux (A B C : Type) (go : gmap_dep_ne A
→ gmap_dep_ne B → gmap_dep C)
(f : option A → option B → option C) (mt1 : gmap_dep A)
(mt2 : gmap_dep B) :=
match mt1 with
| GEmpty =>
match mt2 with
| GEmpty => GEmpty
| GNodes t2' => gmap_dep_ne_omap _ _ (λ x : B, f None (Some x)) t2'
end
| GNodes t1' =>
match mt2 with
| GEmpty => gmap_dep_ne_omap _ _ (λ x : A, f (Some x) None) t1'
| GNodes t2' => go t1' t2'
end
end.
Definition diag_None' {A B C : Type} (f : option A → option B → option C)
(mx : option (A)) (my : option (B)) :=
match mx with
| Some (x) =>
match my with
| Some (y) => f (Some x) (Some y)
| None => f (Some x) None
end
| None =>
match my with
| Some (y) => f None (Some y)
| None => None
end
end.
Definition gmap_dep_ne_merge {A B C : Type} (f : option A → option B → option C) :=
fix go
(t1 : gmap_dep_ne A) (t2 : gmap_dep_ne B)
{struct t1} : gmap_dep C :=
gmap_dep_ne_case _ _ t1
(λ (ml1 : gmap_dep A)
(mx1 : option (A)) (mr1 : gmap_dep A) ,
gmap_dep_ne_case _ _ t2
(λ (ml2 : gmap_dep B)
(mx2 : option (B)) (mr2 : gmap_dep B),
GNode _
(gmap_merge_aux _ _ _ go
f ml1 ml2) (diag_None' f mx1 mx2)
(gmap_merge_aux _ _ _ go
f mr1 mr2))).
Definition gmap_dep_merge {A B C : Type}
(f : option A → option B → option C) :=
gmap_merge_aux _ _ _ (gmap_dep_ne_merge f) f
: gmap_dep A → gmap_dep B → gmap_dep C.
Definition gmap_merge (K : Type) (H : RelDecision (@eq K))
(A B C : Type) (f : option A → option B → option C) := (fun '{| gmap_car := mt1 |} '{| gmap_car := mt2 |} =>
{| gmap_car := gmap_dep_merge f mt1 mt2 |})
: (gmap H A) -> (gmap H B) -> (gmap H C).
Fixpoint gtest_merge {K : Type} (H : RelDecision (@eq K)) (t1 t2 : gtest H) {struct t1} : gtest H :=
match t1, t2 with
| GTest ts1, GTest ts2 =>
GTest (gmap_merge K H _ _ _ (@option_union_with _ (λ t1 t2, Some (gtest_merge H t1 t2))) ts1 ts2)
end.
(* An example from metacoq (simplified) *)
Notation "x .π2" := (projT2 x) (at level 0).
Parameter term : Type.
Inductive All (P : term -> Type) : Type :=
| All_cons : {x:term & P x} -> All P -> All P.
Inductive inferring (Σ : unit) : term -> Type :=
| infer x : All (inferring Σ) -> inferring Σ x.
Fixpoint inferring_size {Σ t} (d : inferring Σ t) {struct d} : nat :=
match d with
| infer _ _ (All_cons _ p _) => inferring_size p.π2
end.
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