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George Pirlea
Iris
Commits
18841bdb
Commit
18841bdb
authored
Jan 20, 2016
by
Robbert Krebbers
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Some more Forall3 lemmas.
And use more uniform variable names.
parent
aea3b304
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prelude/list.v
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18841bdb
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@@ -1372,6 +1372,11 @@ Proof.
induction l as [|x l IH]; [done|].
by rewrite reverse_cons, (commutative (++)), IH.
Qed.
Lemma delete_Permutation l i x : l !! i = Some x → l ≡ₚ x :: delete i l.
Proof.
revert i; induction l as [|y l IH]; intros [|i] ?; simplify_equality'; auto.
by rewrite Permutation_swap, <-(IH i).
Qed.
(** ** Properties of the [prefix_of] and [suffix_of] predicates *)
Global Instance: PreOrder (@prefix_of A).
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@@ -2522,59 +2527,85 @@ 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' :
Lemma Forall3_app l1
l2 k1 k2 k1'
k2' :
Forall3 P l1 k1 k1' → Forall3 P l2 k2 k2' →
Forall3 P (l1 ++ l2) (k1 ++ k2) (k1' ++ k2').
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.
Lemma Forall3_cons_inv_l x l k k' :
Forall3 P (x :: l) k k' → ∃ y k2 z k2',
k = y :: k2 ∧ k' = z :: k2' ∧ P x y z ∧ Forall3 P l k2 k2'.
Proof. inversion_clear 1; naive_solver. Qed.
Lemma Forall3_app_inv_l l1 l2 k k' :
Forall3 P (l1 ++ l2) k k' → ∃ k1 k2 k1' k2',
k = k1 ++ k2 ∧ k' = k1' ++ k2' ∧ Forall3 P l1 k1 k1' ∧ Forall3 P l2 k2 k2'.
Proof.
revert k 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_cons_inv_m l y k k' :
Forall3 P l (y :: k) k' → ∃ x l2 z k2',
l = x :: l2 ∧ k' = z :: k2' ∧ P x y z ∧ Forall3 P l2 k k2'.
Proof. inversion_clear 1; naive_solver. Qed.
Lemma Forall3_app_inv_m l k1 k2 k' :
Forall3 P l (k1 ++ k2) k' → ∃ l1 l2 k1' k2',
l = l1 ++ l2 ∧ k' = k1' ++ k2' ∧ Forall3 P l1 k1 k1' ∧ Forall3 P l2 k2 k2'.
Proof.
revert l k'. induction k1 as [|x k1 IH]; simpl; inversion_clear 1.
* by repeat eexists; eauto.
* by repeat eexists; eauto.
* edestruct IH as (?&?&?&?&?&?&?&?); eauto; naive_solver.
Qed.
Lemma Forall3_cons_inv_r l k z k' :
Forall3 P l k (z :: k') → ∃ x l2 y k2,
l = x :: l2 ∧ k = y :: k2 ∧ P x y z ∧ Forall3 P l2 k2 k'.
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
.
Lemma Forall3_app_inv_
r l k k1' k2'
:
Forall3 P l
k (k1' ++ k2')
→ ∃ l1 l2 k1 k2,
l = l1 ++ l2 ∧ k = k1 ++ k2 ∧ Forall3 P l1
k1 k1' ∧ Forall3 P l2 k2 k2'
.
Proof.
revert l k. induction
l1' as [|x l
1' IH]; simpl; inversion_clear 1.
revert l k. induction
k1' as [|x k
1' 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; auto. Defined.
Lemma Forall3_length_lm l
l' k : Forall3 P l l' k → length l = length l'
.
Lemma Forall3_impl (Q : A → B → C → Prop) l
k k'
:
Forall3 P l
k k' → (∀ x y z, P x y z → Q x y z) → Forall3 Q l k k'
.
Proof. intros Hl ?
;
induction Hl; auto. Defined.
Lemma Forall3_length_lm l
k k' : Forall3 P l k k' → length l = length k
.
Proof. by induction 1; f_equal'. Qed.
Lemma Forall3_length_lr l
l' k : Forall3 P l l' k → length l = length k
.
Lemma Forall3_length_lr l
k k' : Forall3 P l k 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.
Lemma Forall3_lookup_lmr l
k k'
i x y z :
Forall3 P l
k k'
→
l !! i = Some x →
k !! i = Some y → k'
!! i = Some z → P x y z.
Proof.
intros H. revert i. induction H; intros [|?] ???; simplify_equality'; eauto.
Qed.
Lemma Forall3_lookup_l l
l' k
i x :
Forall3 P l
l' k
→ l !! i = Some x →
∃ y z,
l' !! i = Some y ∧ k
!! i = Some z ∧ P x y z.
Lemma Forall3_lookup_l l
k k'
i x :
Forall3 P l
k k'
→ l !! i = Some x →
∃ y z,
k !! i = Some y ∧ k'
!! i = Some z ∧ P x y z.
Proof.
intros H. revert i. induction H; intros [|?] ?; simplify_equality'; eauto.
Qed.
Lemma Forall3_lookup_m l
l' k
i y :
Forall3 P l
l' k → l'
!! i = Some y →
∃ x z, l !! i = Some x ∧ k !! i = Some z ∧ P x y z.
Lemma Forall3_lookup_m l
k k'
i y :
Forall3 P l
k k' → k
!! i = Some y →
∃ x z, l !! i = Some x ∧ k
'
!! i = Some z ∧ P x y z.
Proof.
intros H. revert i. induction H; intros [|?] ?; simplify_equality'; eauto.
Qed.
Lemma Forall3_lookup_r l
l' k
i z :
Forall3 P l
l' k → k
!! i = Some z →
∃ x y, l !! i = Some x ∧
l'
!! i = Some y ∧ P x y z.
Lemma Forall3_lookup_r l
k k'
i z :
Forall3 P l
k k' → k'
!! i = Some z →
∃ x y, l !! i = Some x ∧
k
!! i = Some y ∧ P x y z.
Proof.
intros H. revert i. induction H; intros [|?] ?; simplify_equality'; eauto.
Qed.
Lemma Forall3_alter_lm f g l
l' k
i :
Forall3 P l
l' k
→
(∀ x y z, l !! i = Some x →
l' !! i = Some y → k
!! i = Some z →
Lemma Forall3_alter_lm f g l
k k'
i :
Forall3 P l
k k'
→
(∀ x y z, l !! i = Some x →
k !! 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
.
Forall3 P (alter f i l) (alter g i
k) k'
.
Proof. intros Hl. revert i. induction Hl; intros [|]; auto. Qed.
End Forall3.
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