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From stdpp Require Export relations .
From stdpp Require Import binders gmap .
Lemma if_iff P Q R S :
( P ↔ Q ) →
( R ↔ S ) →
( ( P → R ) ↔ ( Q → S ) ) .
Proof .
naive_solver .
Qed .
Lemma if_iff' P R S :
( P → R ↔ S ) → ( P → R ) ↔ ( P → S ) .
Proof . tauto . Qed .
Lemma and_iff' ( P R S : Prop ) :
( P → R ↔ S ) → ( P ∧ R ) ↔ ( P ∧ S ) .
Proof . tauto . Qed .
Lemma and_iff ( P Q R S : Prop ) :
( P ↔ Q ) → ( ( P ∨ Q ) → R ↔ S ) → ( P ∧ R ) ↔ ( Q ∧ S ) .
Proof . tauto . Qed .
Lemma list_subseteq_cons { X } ( A B : list X ) x : A ⊆ B → x :: A ⊆ x :: B .
Proof . intros Hincl . intros y . rewrite ! elem_of_cons . naive_solver . Qed .
Lemma list_subseteq_cons_binder A B x : A ⊆ B → x : b : A ⊆ x : b : B .
Proof . destruct x ; [ done |] . apply list_subseteq_cons . Qed .
Lemma list_subseteq_cons_l { X } ( A B : list X ) x : A ⊆ x :: B → x :: A ⊆ x :: B .
Proof .
intros Hincl . intros y . rewrite elem_of_cons . intros [ -> | ? ] .
- left .
- apply Hincl . naive_solver .
Qed .
Lemma list_subseteq_cons_elem { X } ( A B : list X ) x :
x ∈ B → A ⊆ B → ( x :: A ) ⊆ B .
Proof .
intros Hel Hincl .
intros a [ -> | ? ] % elem_of_cons ; [ done |] .
by apply Hincl .
Qed .
Lemma elements_subseteq ` { EqDecision A } ` { Countable A } ( X Y : gset A ) :
X ⊆ Y → elements X ⊆ elements Y .
Proof .
rewrite elem_of_subseteq .
intros Ha a . rewrite ! elem_of_elements .
apply Ha .
Qed .
Lemma list_subseteq_cons_r { X } ( A B : list X ) x :
A ⊆ B → A ⊆ ( x :: B ) .
Proof .
intros Hincl . trans B ; [ done |] .
intros b Hel . apply elem_of_cons ; by right .
Qed .