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Simon Spies
stdpp
Commits
18e47df6
Commit
18e47df6
authored
Sep 06, 2014
by
Robbert Krebbers
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Lemmas on Forall3 and tweak list tactics.
parent
93de71b2
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theories/list.v
theories/list.v
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theories/list.v
View file @
18e47df6
...
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@@ -306,7 +306,7 @@ a list equality involving [(::)] and [(++)] of two lists that have a different
length as one of its hypotheses. *)
Tactic Notation "discriminate_list_equality" hyp(H) :=
apply (f_equal length) in H;
repeat (simpl in H || rewrite app_length in H); exfalso; lia.
repeat (
c
simpl in H || rewrite app_length in H); exfalso; lia.
Tactic Notation "discriminate_list_equality" :=
match goal with
| H : @eq (list _) _ _ |- _ => discriminate_list_equality H
...
...
@@ -324,19 +324,16 @@ Proof.
intros ? Hl. apply app_injective_1; auto.
apply (f_equal length) in Hl. rewrite !app_length in Hl. lia.
Qed.
Ltac simplify_list_equality :=
Ltac
simple_
simplify_list_equality :=
repeat match goal with
| _ => progress simplify_equality
| _ => progress simplify_equality
'
| H : _ ++ _ = _ ++ _ |- _ => first
[ apply app_inv_head in H | apply app_inv_tail in H
| apply app_injective_1 in H; [destruct H|done]
| apply app_injective_2 in H; [destruct H|done] ]
| H : [?x] !! ?i = Some ?y |- _ =>
destruct i; [change (Some x = Some y) in H | discriminate]
end;
try discriminate_list_equality.
Ltac simplify_list_equality' :=
repeat (progress simpl in * || simplify_list_equality).
end.
(** * General theorems *)
Section general_properties.
...
...
@@ -1358,7 +1355,7 @@ Proof. by rewrite prefix_suffix_reverse, !reverse_involutive. Qed.
Lemma suffix_of_nil l : [] `suffix_of` l.
Proof. exists l. by rewrite (right_id_L [] (++)). Qed.
Lemma suffix_of_nil_inv l : l `suffix_of` [] → l = [].
Proof. by intros [[|?] ?]; simplify_list_equality. Qed.
Proof. by intros [[|?] ?];
simple_
simplify_list_equality. Qed.
Lemma suffix_of_cons_nil_inv x l : ¬x :: l `suffix_of` [].
Proof. by intros [[] ?]. Qed.
Lemma suffix_of_snoc l1 l2 x :
...
...
@@ -1375,19 +1372,20 @@ Proof. intros ->. apply suffix_of_app. Qed.
Lemma suffix_of_snoc_inv_1 x y l1 l2 :
l1 ++ [x] `suffix_of` l2 ++ [y] → x = y.
Proof.
intros [k' E]. rewrite (associative_L (++)) in E. by simplify_list_equality.
intros [k' E]. rewrite (associative_L (++)) in E.
by simple_simplify_list_equality.
Qed.
Lemma suffix_of_snoc_inv_2 x y l1 l2 :
l1 ++ [x] `suffix_of` l2 ++ [y] → l1 `suffix_of` l2.
Proof.
intros [k' E]. exists k'. rewrite (associative_L (++)) in E.
by simplify_list_equality.
by
simple_
simplify_list_equality.
Qed.
Lemma suffix_of_app_inv l1 l2 k :
l1 ++ k `suffix_of` l2 ++ k → l1 `suffix_of` l2.
Proof.
intros [k' E]. exists k'. rewrite (associative_L (++)) in E.
by simplify_list_equality.
by
simple_
simplify_list_equality.
Qed.
Lemma suffix_of_cons_l l1 l2 x : x :: l1 `suffix_of` l2 → l1 `suffix_of` l2.
Proof. intros [k ->]. exists (k ++ [x]). by rewrite <-(associative_L (++)). Qed.
...
...
@@ -1550,7 +1548,7 @@ Proof.
{ by rewrite <-!(associative_L (++)). }
rewrite sublist_app_l in Hl12. destruct Hl12 as (k1&k2&E&?&Hk2).
destruct k2 as [|z k2] using rev_ind; [inversion Hk2|].
rewrite (associative_L (++)) in E
.
simplify_list_equality.
rewrite (associative_L (++)) in E
; simple_
simplify_list_equality.
eauto using sublist_inserts_r.
Qed.
Global Instance: PartialOrder (@sublist A).
...
...
@@ -2341,15 +2339,31 @@ End Forall2_order.
Section Forall3.
Context {A B C} (P : A → B → C → Prop).
Hint Extern 0 (Forall3 _ _ _ _) => constructor.
Lemma Forall3_app l1 k1 k1' l2 k2 k2' :
Forall3 P l1 k1 k1' → Forall3 P l2 k2 k2' →
Forall3 P (l1 ++ l2) (k1 ++ k2) (k1' ++ k2').
Proof. induction 1; simpl; [done|constructor]; auto. Qed.
Proof. induction 1; simpl; auto. Qed.
Lemma Forall3_cons_inv_m l y l2' k :
Forall3 P l (y :: l2') k → ∃ x l2 z k2,
l = x :: l2 ∧ k = z :: k2 ∧ P x y z ∧ Forall3 P l2 l2' k2.
Proof. inversion_clear 1; naive_solver. Qed.
Lemma Forall3_app_inv_m l l1' l2' k :
Forall3 P l (l1' ++ l2') k → ∃ l1 l2 k1 k2,
l = l1 ++ l2 ∧ k = k1 ++ k2 ∧ Forall3 P l1 l1' k1 ∧ Forall3 P l2 l2' k2.
Proof.
revert l k. induction l1' as [|x l1' IH]; simpl; inversion_clear 1.
* by repeat eexists; eauto.
* by repeat eexists; eauto.
* edestruct IH as (?&?&?&?&?&?&?&?); eauto; naive_solver.
Qed.
Lemma Forall3_impl (Q : A → B → C → Prop) l l' k :
Forall3 P l l' k → (∀ x y z, P x y z → Q x y z) → Forall3 Q l l' k.
Proof. intros Hl ?. induction Hl;
constructor;
auto. Defined.
Proof. intros Hl ?. induction Hl; auto. Defined.
Lemma Forall3_length_lm l l' k : Forall3 P l l' k → length l = length l'.
Proof. by induction 1; f_equal'. Qed.
Lemma Forall3_length_lr l l' k : Forall3 P l l' k → length l = length k.
Proof. by induction 1; f_equal'. Qed.
Lemma Forall3_lookup_lmr l l' k i x y z :
Forall3 P l l' k →
l !! i = Some x → l' !! i = Some y → k !! i = Some z → P x y z.
...
...
@@ -2379,7 +2393,7 @@ Section Forall3.
(∀ x y z, l !! i = Some x → l' !! i = Some y → k !! i = Some z →
P x y z → P (f x) (g y) z) →
Forall3 P (alter f i l) (alter g i l') k.
Proof. intros Hl. revert i. induction Hl; intros [|];
constructor;
auto. Qed.
Proof. intros Hl. revert i. induction Hl; intros [|]; auto. Qed.
End Forall3.
(** * Properties of the monadic operations *)
...
...
@@ -3080,9 +3094,9 @@ Ltac decompose_elem_of_list := repeat
| H : _ ∈ _ :: _ |- _ => apply elem_of_cons in H; destruct H
| H : _ ∈ _ ++ _ |- _ => apply elem_of_app in H; destruct H
end.
Ltac simplify_list_
fmap_
equality := repeat
Ltac simplify_list_equality := repeat
match goal with
| _ => progress simpl
ify
_equality
| _ => progress simpl
e_simplify_list
_equality
| H : _ <$> _ = [] |- _ => apply fmap_nil_inv in H
| H : [] = _ <$> _ |- _ => symmetry in H; apply fmap_nil_inv in H
| H : _ <$> _ = _ :: _ |- _ =>
...
...
@@ -3091,10 +3105,6 @@ Ltac simplify_list_fmap_equality := repeat
| H : _ <$> _ = _ ++ _ |- _ =>
apply fmap_app_inv in H; destruct H as (?&?&?&?&?)
| H : _ ++ _ = _ <$> _ |- _ => symmetry in H
end.
Ltac simplify_zip_equality := repeat
match goal with
| _ => progress simplify_equality
| H : zip_with _ _ _ = [] |- _ => apply zip_with_nil_inv in H; destruct H
| H : [] = zip_with _ _ _ |- _ => symmetry in H
| H : zip_with _ _ _ = _ :: _ |- _ =>
...
...
@@ -3116,15 +3126,17 @@ Ltac decompose_Forall_hyps := repeat
| H : Forall2 _ ?l [] |- _ => apply Forall2_nil_inv_r in H; subst l
| H : Forall2 _ (_ :: _) (_ :: _) |- _ =>
apply Forall2_cons_inv in H; destruct H
| _ => progress simplify_equality'
| H : Forall2 _ (_ :: _) ?k |- _ =>
let k_hd := fresh k "_hd" in let k_tl := fresh k "_tl" in
apply Forall2_cons_inv_l in H; destruct H as (k_hd&k_tl&?&?&->);
rename k_tl into k
| H : Forall2 _ ?l (_ :: _) |- _ =>
let l_hd := fresh l "_hd" in let l_tl := fresh l "_tl" in
apply Forall2_cons_inv_r in H; destruct H as (l_hd&l_tl&?&?&->);
rename l_tl into l
| _ => progress simplify_list_equality
| H : Forall2 _ (_ :: _) ?k |- _ => first
[ let k_hd := fresh k "_hd" in let k_tl := fresh k "_tl" in
apply Forall2_cons_inv_l in H; destruct H as (k_hd&k_tl&?&?&->);
rename k_tl into k
| apply Forall2_cons_inv_l in H; destruct H as (?&?&?&?&?)]
| H : Forall2 _ ?l (_ :: _) |- _ => first
[ let l_hd := fresh l "_hd" in let l_tl := fresh l "_tl" in
apply Forall2_cons_inv_r in H; destruct H as (l_hd&l_tl&?&?&->);
rename l_tl into l
| apply Forall2_cons_inv_r in H; destruct H as (?&?&?&?&?)]
| H : Forall2 _ (_ ++ _) (_ ++ _) |- _ =>
apply Forall2_app_inv in H; [destruct H | first
[by eauto using Forall2_length | by symmetry; eauto using Forall2_length]]
...
...
@@ -3136,6 +3148,10 @@ Ltac decompose_Forall_hyps := repeat
[ let l1 := fresh l "_1" in let l2 := fresh l "_2" in
apply Forall2_app_inv_r in H; destruct H as (l1&l2&?&?&->)
| apply Forall2_app_inv_r in H; destruct H as (?&?&?&?&?) ]
| H : Forall3 _ _ (_ :: _) _ |- _ =>
apply Forall3_cons_inv_m in H; destruct H as (?&?&?&?&?&?&?&?)
| H : Forall3 _ _ (_ ++ _) _ |- _ =>
apply Forall3_app_inv_m in H; destruct H as (?&?&?&?&?&?&?&?)
| H : Forall ?P ?l, H1 : ?l !! _ = Some ?x |- _ =>
(* to avoid some stupid loops, not fool proof *)
unless (P x) by auto using Forall_app_2, Forall_nil_2;
...
...
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