Lvc.Infra.SizeInduction
Disclaimer: This file was taken from the Autosubst library.
https://www.ps.uni-saarland.de/autosubst/
Require Import Coq.Program.Tactics Util Get.
Class Size (A : Type) := size : A → nat.
Ltac gen_Size :=
hnf; match goal with [ |- ?A → nat] ⇒
fix size' 1; intros s;
assert(size_inst : Size A);[exact size' | idtac];
destruct s eqn:E;
let term := type of s in
match goal with
[E : s = ?s' |- _] ⇒
let rec map s :=
(match s with
?s1 ?s2 ⇒
let size_s1 := map s1 in
let s2_T := type of s2 in
let size_s2 := match s2_T with
| A ⇒ constr:(size' s2)
| _ ⇒ constr:(size s2)
end in
constr:(size_s1 + size_s2)
| _ ⇒ constr:(O) end) in
let t' := map s' in
let t'' := eval simpl plus in t' in
exact (S t'')
end
end.
Require Import Omega.
Lemma size_rec {A : Type} f (x : A) : ∀ P : A → Type, (∀ x, (∀ y, f y < f x → P y) → P x) → P x.
Proof.
intros P IS. cut (∀ n x, f x ≤ n → P x). { eauto. }
intros n. induction n; intros; apply IS; intros.
- omega.
- apply IHn. omega.
Defined.
Ltac autorevert x :=
try (match goal with
| [y : ?Y |- ?claim] ⇒
try(match x with y ⇒ idtac end; fail 1);
match goal with [z : _ |- _] ⇒
match claim with appcontext[z] ⇒
first[
match Y with appcontext[z] ⇒ revert y; autorevert x end
| match y with z ⇒ revert y; autorevert x end]
end
end
end).
Tactic Notation "sinduction" ident(H) "using" open_constr(f) :=
let T := type of H in
autorevert H; induction H using (@size_rec _ f);
match goal with [H : T |- _] ⇒ destruct H end.
Tactic Notation "sinduction" ident(H) :=
let T := type of H in
autorevert H; induction H using (@size_rec _ size);
match goal with [H : T |- _] ⇒ destruct H end; intros.
Ltac sind H :=
let IH := fresh "IH" in
let x := fresh "x" in
induction H as [x IH] using (@size_rec _ size); try rename x into H.
Instance def_size (A : Type) : Size A | 100 := (fun x ⇒ O).
Instance size_nat : Size nat := id.
Instance size_prod {A B : Type} (size_A : Size A) (size_B : Size B)
: Size (prod A B).
intros [a b]. eapply (size a + size b).
Defined.
Instance size_list (A : Type) (size_A : Size A) : Size (list A). gen_Size. Defined.
Lemma size_list_get {A : Type} `{Size A} {L : list A} {n} {a}
: get L n a → size a ≤ size L.
Proof.
general induction L; inv H0; repeat(unfold size in *; unfold def_size in *; simpl in *).
- omega.
- rewrite IHL; eauto. omega.
Qed.
Lemma size_prod_split {A B : Type} `{Size A} `{Size B} {x:A×B} {n}
: size_prod _ _ x ≤ n → size (fst x) ≤ n ∧ size (snd x) ≤ n.
Proof.
destruct x; simpl; intros. omega.
Qed.
Tactics
Ltac sizesimpl :=
repeat (unfold size in × |- *; unfold def_size in × |- *; simpl in × |- *;
try match goal with
| [ H: get ?L ?n ?x |- context [size_list _ _ ?L] ]
⇒ eapply size_list_get in H
| [ H: size_prod _ _ _ ≤ _ |- _ ]
⇒ eapply size_prod_split in H
end; clear_trivial_eqs).
Tactic Notation "somega" := sizesimpl; omega.
Hint Extern 2 (size _ < size _) ⇒ somega.
Hint Extern 2 (size _ < S (size _)) ⇒ somega.
Hint Extern 2 (S (size _) < S (size _)) ⇒ somega.
Tests