Require Import Vbase NPeano Omega Permutation List Relations Setoid Classical.
Set Implicit Arguments.
Lemma list_seq_split :
∀ x a y,
x ≤ y →
List.seq a y = List.seq a x ++ List.seq (x + a) (y - x).
Require Import extralib.
Definitions of relations
Make arguments implicit
Definition immediate X (rel : relation X) (a b: X) :=
rel a b ∧ (∀ c (R1: rel a c) (R2: rel c b), False).
Definition irreflexive X (rel : relation X) := ∀ x, rel x x → False.
Definition acyclic X (rel : relation X) := irreflexive (clos_trans rel).
Definition is_total X (cond: X → Prop) (rel: relation X) :=
∀ a (IWa: cond a)
b (IWb: cond b) (NEQ: a ≠ b),
rel a b ∨ rel b a.
Definition restr_subset X (cond: X → Prop) (rel rel': relation X) :=
∀ a (IWa: cond a)
b (IWb: cond b) (REL: rel a b),
rel' a b.
Definition restr_rel X (cond : X → Prop) (rel : relation X) : relation X :=
fun a b ⇒ rel a b ∧ cond a ∧ cond b.
Definition restr_eq_rel A B (f : A → B) rel x y :=
rel x y ∧ f x = f y.
Definition upward_closed (X: Type) (rel: relation X) (P: X → Prop) :=
∀ x y (REL: rel x y) (POST: P y), P x.
Definition max_elt (X: Type) (rel: relation X) (a: X) :=
∀ b (REL: rel a b), False.
Notation "r 'UNION1' ( a , b )" :=
(fun x y ⇒ x = a ∧ y = b ∨ r x y) (at level 100).
Notation "a <--> b" := (same_relation a b) (at level 110).
Definition seq (X:Type) (r1 r2 : relation X) : relation X :=
fun x y ⇒ ∃ z, r1 x z ∧ r2 z y.
Definition clos_refl A (R: relation A) x y := x = y ∨ R x y.
Definition eqv_rel A f (x y : A) := x = y ∧ f x.
Notation "P +++ Q" := (union P Q) (at level 50, left associativity).
Notation "P ;; Q" := (seq P Q) (at level 45, right associativity).
Very basic properties of relations
Lemma r_refl A (R: relation A) x : clos_refl R x x.
Lemma r_step A (R: relation A) x y : R x y → clos_refl R x y.
Hint Immediate r_refl r_step.
Section BasicProperties.
Variable A : Type.
Variable dom : A → Prop.
Variables r r' r'' : relation A.
Lemma clos_trans_mon a b :
clos_trans r a b →
(∀ a b, r a b → r' a b) →
clos_trans r' a b.
Lemma clos_refl_trans_mon a b :
clos_refl_trans r a b →
(∀ a b, r a b → r' a b) →
clos_refl_trans r' a b.
Lemma clos_refl_transE a b :
clos_refl_trans r a b ↔ a = b ∨ clos_trans r a b.
Lemma clos_trans_in_rt a b :
clos_trans r a b → clos_refl_trans r a b.
Lemma rt_t_trans a b c :
clos_refl_trans r a b → clos_trans r b c → clos_trans r a c.
Lemma t_rt_trans a b c :
clos_trans r a b → clos_refl_trans r b c → clos_trans r a c.
Lemma t_step_rt x y :
clos_trans r x y ↔ ∃ z, r x z ∧ clos_refl_trans r z y.
Lemma t_rt_step x y :
clos_trans r x y ↔ ∃ z, clos_refl_trans r x z ∧ r z y.
Lemma clos_trans_of_transitive (T: transitive r) x y :
clos_trans r x y → r x y.
Lemma ct_of_transitive (T: transitive r) x y :
clos_trans r x y ↔ r x y.
Lemma crt_of_transitive (T: transitive r) x y :
clos_refl_trans r x y ↔ clos_refl r x y.
Lemma clos_trans_eq :
∀ B (f : A → B)
(H: ∀ a b (SB: r a b), f a = f b) a b
(C: clos_trans r a b),
f a = f b.
Lemma trans_irr_acyclic :
irreflexive r → transitive r → acyclic r.
Lemma restr_rel_trans :
transitive r → transitive (restr_rel dom r).
Lemma upward_clos_trans P :
upward_closed r P → upward_closed (clos_trans r) P.
Lemma max_elt_clos_trans a b:
max_elt r a → clos_trans r a b → False.
Lemma is_total_restr :
is_total dom r →
is_total dom (restr_rel dom r).
Lemma clos_trans_restrD f x y :
clos_trans (restr_rel f r) x y → f x ∧ f y.
Lemma clos_trans_restr_eqD B (f: A → B) x y :
clos_trans (restr_eq_rel f r) x y → f x = f y.
Lemma irreflexive_inclusion:
inclusion r r' →
irreflexive r' →
irreflexive r.
Lemma irreflexive_union :
irreflexive (union r r') ↔ irreflexive r ∧ irreflexive r'.
Lemma irreflexive_seqC :
irreflexive (seq r r') ↔ irreflexive (seq r' r).
Lemma clos_trans_inclusion :
inclusion r (clos_trans r).
Lemma clos_trans_inclusion_clos_refl_trans:
inclusion (clos_trans r) (clos_refl_trans r).
Lemma clos_trans_monotonic :
inclusion r r' →
inclusion (clos_trans r) (clos_trans r').
Lemma inclusion_seq_mon s s' :
inclusion r r' →
inclusion s s' →
inclusion (r ;; s) (r' ;; s').
Lemma inclusion_seq_trans t :
transitive t →
inclusion r t →
inclusion r' t →
inclusion (seq r r') t.
Lemma inclusion_seq_rt :
inclusion r (clos_refl_trans r'') →
inclusion r' (clos_refl_trans r'') →
inclusion (seq r r') (clos_refl_trans r'').
Lemma inclusion_union_l :
inclusion r r'' →
inclusion r' r'' →
inclusion (union r r') r''.
Lemma inclusion_union_r :
inclusion r r' ∨ inclusion r r'' →
inclusion r (union r' r'').
Lemma inclusion_step_rt :
inclusion r r' →
inclusion r (clos_refl_trans r').
Lemma inclusion_r_rt :
inclusion r r' →
inclusion (clos_refl r) (clos_refl_trans r').
Lemma inclusion_rt_rt :
inclusion r r' →
inclusion (clos_refl_trans r) (clos_refl_trans r').
Lemma inclusion_step_t :
inclusion r r' →
inclusion r (clos_trans r').
Lemma inclusion_t_t :
inclusion r r' →
inclusion (clos_trans r) (clos_trans r').
Lemma inclusion_acyclic :
inclusion r r' →
acyclic r' →
acyclic r.
Lemma irreflexive_restr :
irreflexive r → irreflexive (restr_rel dom r).
Lemma inclusion_restr :
inclusion (restr_rel dom r) r.
End BasicProperties.
Lemma transitive_restr_eq A B (f: A → B) r :
transitive r → transitive (restr_eq_rel f r).
Lemma irreflexive_restr_eq A B (f: A → B) r :
irreflexive (restr_eq_rel f r) ↔ irreflexive r.
Lemma clos_trans_of_clos_trans1 A (r r' : relation A) x y :
clos_trans (fun a b ⇒ clos_trans r a b ∨ r' a b) x y ↔
clos_trans (fun a b ⇒ r a b ∨ r' a b) x y.
Lemma clos_trans_of_clos_trans A (r : relation A) x y :
clos_trans (clos_trans r) x y ↔
clos_trans r x y.
Lemma inclusion_union :
∀ {A : Type} (R1 R1' R2 R2' : relation A)
(HINC1 : inclusion R1' R1)
(HINC2 : inclusion R2' R2),
inclusion (union R2' R1') (union R2 R1).
Lemma inclusion_restr_rel_l :
∀ {A : Type} (dom : A → Prop) (R1 R1' : relation A)
(HINC: inclusion R1' R1),
inclusion (restr_rel dom R1') R1.
Lemma inclusion_seq_refl :
∀ (A : Type) (R1 R2 R3 : relation A)
(INC1: inclusion R1 R3)
(INC2: inclusion R2 R3)
(TRANS: transitive R3),
inclusion (seq R1 (clos_refl R2)) R3.
Lemma inclusion_rt_l X (r r' : relation X) :
reflexive r' →
inclusion (seq r r') r' →
inclusion (clos_refl_trans r) r'.
Lemma inclusion_rt_rt2 :
∀ (A : Type) (r t : relation A),
inclusion r (clos_refl_trans t) →
inclusion (clos_refl_trans r) (clos_refl_trans t).
Lemma inclusion_restr_eq A B (f: A → B) r :
inclusion (restr_eq_rel f r) r.
Hint Resolve
inclusion_restr_eq inclusion_restr
inclusion_acyclic inclusion_restr_rel_l
inclusion_step_t inclusion_union_r inclusion_union
inclusion_seq_refl : inclusion.
Hint Resolve inclusion_rt_rt inclusion_r_rt inclusion_step_rt : inclusion.
Set up setoid rewriting
First, for inclusion.
Lemma inclusion_refl A : reflexive (@inclusion A).
Lemma inclusion_trans A : transitive (@inclusion A).
Add Parametric Relation (X : Type) : (relation X) (@inclusion X)
reflexivity proved by (@inclusion_refl X)
transitivity proved by (@inclusion_trans X)
as inclusion_rel.
Add Parametric Morphism X : (@inclusion X) with signature
inclusion --> inclusion ++> Basics.impl as inclusion_mori.
Add Parametric Morphism X : (@union X) with signature
inclusion ==> inclusion ==> inclusion as union_mori.
Add Parametric Morphism X : (@seq X) with signature
inclusion ==> inclusion ==> inclusion as seq_mori.
Add Parametric Morphism X : (@irreflexive X) with signature
inclusion --> Basics.impl as irreflexive_mori.
Add Parametric Morphism X : (@clos_trans X) with signature
inclusion ==> inclusion as clos_trans_mori.
Add Parametric Morphism X : (@clos_refl_trans X) with signature
inclusion ==> inclusion as clos_refl_trans_mori.
Add Parametric Morphism X : (@clos_refl X) with signature
inclusion ==> inclusion as clos_refl_mori.
Add Parametric Morphism X P : (@restr_rel X P) with signature
inclusion ==> inclusion as restr_rel_mori.
Add Parametric Morphism X : (@acyclic X) with signature
inclusion --> Basics.impl as acyclic_mori.
Add Parametric Morphism X : (@is_total X) with signature
eq ==> inclusion ==> Basics.impl as is_total_mori.
Second, for equivalence.
Lemma same_relation_exp A (r r' : relation A) (EQ: r <--> r') :
∀ x y, r x y ↔ r' x y.
Lemma same_relation_refl A : reflexive (@same_relation A).
Lemma same_relation_sym A : symmetric (@same_relation A).
Lemma same_relation_trans A : transitive (@same_relation A).
Add Parametric Relation (X : Type) : (relation X) (@same_relation X)
reflexivity proved by (@same_relation_refl X)
symmetry proved by (@same_relation_sym X)
transitivity proved by (@same_relation_trans X)
as same_rel.
Add Parametric Morphism X : (@inclusion X) with signature
same_relation ==> same_relation ==> iff as inclusion_more.
Add Parametric Morphism X : (@union X) with signature
same_relation ==> same_relation ==> same_relation as union_more.
Add Parametric Morphism X : (@seq X) with signature
same_relation ==> same_relation ==> same_relation as seq_more.
Add Parametric Morphism X P : (@restr_rel X P) with signature
same_relation ==> same_relation as restr_rel_more.
Add Parametric Morphism X : (@clos_trans X) with signature
same_relation ==> same_relation as clos_trans_more.
Add Parametric Morphism X : (@clos_refl_trans X) with signature
same_relation ==> same_relation as clos_relf_trans_more.
Add Parametric Morphism X : (@clos_refl X) with signature
same_relation ==> same_relation as clos_relf_more.
Add Parametric Morphism X : (@irreflexive X) with signature
same_relation ==> iff as irreflexive_more.
Add Parametric Morphism X : (@acyclic X) with signature
same_relation ==> iff as acyclic_more.
Add Parametric Morphism X : (@transitive X) with signature
same_relation ==> iff as transitive_more.
Add Parametric Morphism X : (@clos_trans X) with signature
same_relation ==> eq ==> eq ==> iff as clos_trans_more'.
Add Parametric Morphism X : (@clos_refl_trans X) with signature
same_relation ==> eq ==> eq ==> iff as clos_refl_trans_more'.
Add Parametric Morphism X : (@is_total X) with signature
eq ==> same_relation ==> iff as is_total_more.
Lemma same_relation_restr X (f : X → Prop) rel rel' :
(∀ x (CONDx: f x) y (CONDy: f y), rel x y ↔ rel' x y) →
(restr_rel f rel <--> restr_rel f rel').
Lemma union_restr X (f : X → Prop) rel rel' :
union (restr_rel f rel) (restr_rel f rel')
<--> restr_rel f (union rel rel').
Lemma clos_trans_restr X (f : X → Prop) rel (UC: upward_closed rel f) :
clos_trans (restr_rel f rel)
<--> restr_rel f (clos_trans rel).
Lemma seq_union_l X (r1 r2 r : relation X) :
seq (union r1 r2) r <--> union (seq r1 r) (seq r2 r).
Lemma seq_union_r X (r r1 r2 : relation X) :
seq r (union r1 r2) <--> union (seq r r1) (seq r r2).
Lemma seqA X (r1 r2 r3 : relation X) :
seq (seq r1 r2) r3 <--> seq r1 (seq r2 r3).
Lemma unionA X (r1 r2 r3 : relation X) :
union (union r1 r2) r3 <--> union r1 (union r2 r3).
Lemma unionC X (r1 r2 : relation X) :
union r1 r2 <--> union r2 r1.
Lemma rtE_left X (r r' : relation X) :
r ;; clos_refl_trans r' <--> r +++ ((r ;; r') ;; clos_refl_trans r').
Lemma rtE_right X (r r' : relation X) :
clos_refl_trans r' ;; r <--> r +++ (clos_refl_trans r' ;; r' ;; r).
Lemma t_step_rt2 X (r : relation X) :
clos_trans r <--> r ;; clos_refl_trans r.
Lemma seqFr X (r : relation X) :
(fun _ _ ⇒ False) ;; r <--> (fun _ _ ⇒ False).
Lemma seqrF X (r : relation X) :
r ;; (fun _ _ ⇒ False) <--> (fun _ _ ⇒ False).
Lemma unionrF X (r : relation X) :
r +++ (fun _ _ ⇒ False) <--> r.
Lemma unionFr X (r : relation X) :
(fun _ _ ⇒ False) +++ r <--> r.
Lemma crt_seq_swap X (r r' : relation X) :
clos_refl_trans (r ;; r') ;; r <-->
r ;; clos_refl_trans (r' ;; r).
Lemma crt_double X (r : relation X) :
clos_refl_trans r <-->
clos_refl r ;; clos_refl_trans (r ;; r).
Hint Rewrite seqFr seqrF unionrF unionFr : samerel.
Lemma min_cycle X (rel rel' : relation X) (dom : X → Prop)
(TOT: is_total dom rel')
(T : transitive rel')
(INCL: inclusion rel' (clos_trans rel))
(INV: ∀ a b (R: rel a b) (R': rel' b a), False) :
acyclic rel ↔
acyclic (restr_rel (fun x ⇒ ¬ dom x) rel) ∧
(∀ x (CYC: rel x x) (D: dom x), False) ∧
(∀ c1 b1 (R: rel c1 b1) b2
(S : clos_refl rel' b1 b2) c2
(R': rel b2 c2) (S': clos_refl_trans (restr_rel (fun x ⇒ ¬ dom x) rel) c2 c1)
(D1 : dom b1) (D2: dom b2) (ND1: ¬ dom c1) (ND2: ¬ dom c2), False).
Lemma path_decomp_u1 X (rel : relation X) a b c d :
clos_trans (rel UNION1 (a, b)) c d →
clos_trans rel c d ∨
clos_refl_trans rel c a ∧ clos_refl_trans rel b d.
Lemma cycle_decomp_u1 X (rel : relation X) a b c :
clos_trans (rel UNION1 (a, b)) c c →
clos_trans rel c c ∨ clos_refl_trans rel b a.
Lemma path_decomp_u_total :
∀ X (rel1 : relation X) dom rel2 (T: is_total dom rel2)
(D: ∀ a b (REL: rel2 a b), dom a ∧ dom b) x y
(C: clos_trans (fun a b ⇒ rel1 a b ∨ rel2 a b) x y),
clos_trans rel1 x y ∨
(∃ m n,
clos_refl_trans rel1 x m ∧ clos_trans rel2 m n ∧ clos_refl_trans rel1 n y) ∨
(∃ m n,
clos_refl_trans rel1 m n ∧ clos_trans rel2 n m).
Lemma cycle_decomp_u_total :
∀ X (rel1 : relation X) dom rel2 (T: is_total dom rel2)
(D: ∀ a b (REL: rel2 a b), dom a ∧ dom b) x
(C: clos_trans (fun a b ⇒ rel1 a b ∨ rel2 a b) x x),
clos_trans rel1 x x ∨
(∃ m n, clos_refl_trans rel1 m n ∧ clos_trans rel2 n m).
Lemma clos_trans_disj_rel :
∀ X (rel rel' : relation X)
(DISJ: ∀ x y (R: rel x y) z (R': rel' y z), False) x y
(R: clos_trans rel x y) z
(R': clos_trans rel' y z),
False.
Lemma path_decomp_u_1 :
∀ X (rel rel' : relation X)
(DISJ: ∀ x y (R: rel x y) z (R': rel' y z), False) x y
(T: clos_trans (union rel rel') x y),
clos_trans rel x y ∨ clos_trans rel' x y
∨ ∃ z, clos_trans rel' x z ∧ clos_trans rel z y.
Lemma cycle_decomp_u_1 :
∀ X (rel rel' : relation X)
(DISJ: ∀ x y (R: rel x y) z (R': rel' y z), False) x
(T: clos_trans (union rel rel') x x),
clos_trans rel x x ∨ clos_trans rel' x x.
Lemma cycle_disj :
∀ X (rel : relation X)
(DISJ: ∀ x y (R: rel x y) z (R': rel y z), False) x
(T: clos_trans rel x x), False.
Lemma clos_trans_restr_trans_mid :
∀ X (rel rel' : relation X) f x y
(A : clos_trans (restr_rel f (fun x y ⇒ rel x y ∨ rel' x y)) x y)
z (B : rel y z) w
(C : clos_trans (restr_rel f (fun x y ⇒ rel x y ∨ rel' x y)) z w),
clos_trans (restr_rel f (fun x y ⇒ rel x y ∨ rel' x y)) x w.
Lemma clos_trans_restr_trans_cycle :
∀ X (rel rel' : relation X) f x y
(A : clos_trans (restr_rel f (fun x y ⇒ rel x y ∨ rel' x y)) x y)
(B : rel y x),
clos_trans (restr_rel f (fun x y ⇒ rel x y ∨ rel' x y)) x x.
Lemma path_tur :
∀ X (r r' : relation X) (adom bdom : X → Prop)
(T: transitive r)
(A: ∀ x y (R: r' x y), adom x)
(B: ∀ x y (R: r' x y), bdom y) x y
(P: clos_trans (fun x y ⇒ r x y ∨ r' x y) x y),
r x y ∨
∃ z,
clos_trans (fun x y ⇒ r x y ∧ adom y ∨ r' x y) x z ∧
(z = y ∨ r z y ∧ bdom z).
Lemma path_ur :
∀ X (r r' : relation X) (adom bdom : X → Prop)
(A: ∀ x y (R: r' x y), adom x)
(B: ∀ x y (R: r' x y), bdom y) x y
(P: clos_trans (fun x y ⇒ r x y ∨ r' x y) x y),
clos_trans r x y ∨
∃ z,
clos_trans (fun x y ⇒ clos_trans r x y ∧ adom y ∨ r' x y) x z ∧
(z = y ∨ clos_trans r z y ∧ bdom z).
Lemma path_tur2 :
∀ X (r r' : relation X) (adom bdom : X → Prop)
(T: transitive r')
(A: ∀ x y (R: r x y), adom x)
(B: ∀ x y (R: r x y), bdom y) x y
(P: clos_trans (fun x y ⇒ r x y ∨ r' x y) x y),
r' x y ∨
∃ z,
(x = z ∨ r' x z ∧ adom z) ∧
clos_trans (fun x y ⇒ r x y ∨ r' x y ∧ bdom x) z y.
Lemma path_ur2 :
∀ X (r r' : relation X) (adom bdom : X → Prop)
(A: ∀ x y (R: r x y), adom x)
(B: ∀ x y (R: r x y), bdom y) x y
(P: clos_trans (fun x y ⇒ r x y ∨ r' x y) x y),
clos_trans r' x y ∨
∃ z,
(x = z ∨ clos_trans r' x z ∧ adom z) ∧
clos_trans (fun x y ⇒ r x y ∨ clos_trans r' x y ∧ bdom x) z y.
Lemma cycle_ur :
∀ X (r r' : relation X) (adom bdom : X → Prop)
(A: ∀ x y (R: r x y), adom x)
(B: ∀ x y (R: r x y), bdom y) x
(P: clos_trans (fun x y ⇒ r x y ∨ r' x y) x x),
clos_trans r' x x ∨
∃ x,
clos_trans (fun x y ⇒ r x y ∨ clos_trans r' x y ∧ bdom x ∧ adom y) x x.
Lemma restr_eq_union :
∀ X (rel rel' : relation X) B (f: X → B) x y
(R: ∀ x y, rel' x y → f x = f y),
restr_eq_rel f (fun x y ⇒ rel x y ∨ rel' x y) x y ↔
restr_eq_rel f rel x y ∨ rel' x y.
Lemma clos_trans_restr_eq_union :
∀ X (rel rel' : relation X) B (f: X → B)
(R: ∀ x y, rel' x y → f x = f y),
clos_trans (restr_eq_rel f (fun x y ⇒ rel x y ∨ rel' x y)) <-->
clos_trans (fun x y ⇒ restr_eq_rel f rel x y ∨ rel' x y).
Lemma acyclic_mon X (rel rel' : relation X) :
acyclic rel → inclusion rel' rel → acyclic rel'.
Extension of a partial order to a total order
Section one_extension.
Variable X : Type.
Variable elem : X.
Variable rel : relation X.
Definition one_ext : relation X :=
fun x y ⇒
clos_trans rel x y
∨ clos_refl_trans rel x elem ∧ ¬ clos_refl_trans rel y elem.
Lemma one_ext_extends x y : rel x y → one_ext x y.
Lemma one_ext_trans : transitive one_ext.
Lemma one_ext_irr : acyclic rel → irreflexive one_ext.
Lemma one_ext_total_elem :
∀ x, x ≠ elem → one_ext elem x ∨ one_ext x elem.
End one_extension.
Fixpoint tot_ext X (dom : list X) (rel : relation X) : relation X :=
match dom with
| nil ⇒ clos_trans rel
| x::l ⇒ one_ext x (tot_ext l rel)
end.
Lemma tot_ext_extends :
∀ X dom (rel : relation X) x y, rel x y → tot_ext dom rel x y.
Lemma tot_ext_trans X dom (rel : relation X) : transitive (tot_ext dom rel).
Lemma tot_ext_irr :
∀ X (dom : list X) rel, acyclic rel → irreflexive (tot_ext dom rel).
Lemma tot_ext_total :
∀ X (dom : list X) rel, is_total (fun x ⇒ In x dom) (tot_ext dom rel).
Lemma tot_ext_inv :
∀ X dom rel (x y : X),
acyclic rel → tot_ext dom rel x y → ¬ rel y x.
Lemma tot_ext_extends_dom
X dom dom' (rel : relation X) x y :
tot_ext dom rel x y →
tot_ext (dom' ++ dom) rel x y.
Definition tot_ext_nat rel (x y: nat) :=
∃ k, tot_ext (rev (List.seq 0 k)) rel x y.
Lemma tot_ext_nat_extends (rel : relation nat) x y :
rel x y → tot_ext_nat rel x y.
Lemma tot_ext_nat_trans rel : transitive (tot_ext_nat rel).
Lemma tot_ext_nat_irr :
∀ rel, acyclic rel → irreflexive (tot_ext_nat rel).
Lemma tot_ext_nat_total :
∀ rel, is_total (fun _ ⇒ true) (tot_ext_nat rel).
Lemma tot_ext_nat_inv :
∀ rel x y,
acyclic rel → tot_ext_nat rel x y → ¬ rel y x.
Add Parametric Morphism X : (@one_ext X) with signature
eq ==> same_relation ==> same_relation as one_ext_more.
Add Parametric Morphism X : (@tot_ext X) with signature
eq ==> same_relation ==> same_relation as tot_ext_more.
Add Parametric Morphism : tot_ext_nat with signature
same_relation ==> same_relation as tot_ext_nat_more.
Misc properties
Lemma clos_trans_imm :
∀ X (R : relation X) (I: irreflexive R)
(T: transitive R) L (ND: NoDup L) a b
(D: ∀ c, R a c → R c b → In c L)
(REL: R a b),
clos_trans (immediate R) a b.
Lemma clos_trans_rotl A (r r' : relation A) :
clos_trans (r ;; r') <--> r ;; clos_refl_trans (r' ;; r) ;; r'.
Lemma acyclic_rotl A (r r' : relation A) :
acyclic (r ;; r') ↔ acyclic (r' ;; r).
Lemma immediate_clos_trans_elim A (r : relation A) a b :
immediate (clos_trans r) a b →
r a b ∧ (∀ c, clos_trans r a c → clos_trans r c b → False).
Lemma clos_trans_immediate1 A (r : relation A) (T: transitive r) a b :
clos_trans (immediate r) a b → r a b.
Lemma clos_trans_immediate2 A (r : relation A)
(T: transitive r) (IRR: irreflexive r) dom
(D: ∀ a b (R: r a b), In b dom) a b :
r a b →
clos_trans (immediate r) a b.
Preferential union
Definition pref_union X (r r' : relation X) x y :=
r x y ∨ r' x y ∧ ¬ r y x.
Lemma acyclic_pref_union :
∀ X (r r' : relation X) (dom : X → Prop)
(IRR: irreflexive r)
(T: transitive r)
(TOT: is_total dom r)
(DL: ∀ x y (R: r' x y), dom x ∧ ¬ dom y),
acyclic (pref_union r r').
Lemma in_pref_union :
∀ X (r r' : relation X) (dom : X → Prop)
(IRR: irreflexive r)
(T: transitive r)
(TOT: is_total dom r)
(DL: ∀ x y (R: r' x y), dom x ∧ ¬ dom y) x y
(R: clos_trans (pref_union r r') x y)
(D: dom y),
r x y.
Remove duplicate list elements (classical)
Fixpoint undup A dec (l: list A) :=
match l with nil ⇒ nil
| x :: l ⇒
if In_dec dec x l then undup dec l else x :: undup dec l
end.
Lemma In_undup X dec (x: X) l : In x (undup dec l) ↔ In x l.
Lemma NoDup_undup X dec (l : list X) : NoDup (undup dec l).
Lemma clos_trans_imm2 :
∀ X (dec : ∀ x y : X, {x = y} + {x ≠ y})
(R : relation X) (I: irreflexive R)
(T: transitive R) L a b
(D: ∀ c, R a c → R c b → In c L)
(REL: R a b),
clos_trans (immediate R) a b.
Lemma total_immediate_unique:
∀ X (eq_X_dec: ∀ (x y: X), {x=y} + {x≠y}) (rel: X → X → Prop) (P: X → Prop)
(Tot: is_total P rel)
a b c (pa: P a) (pb: P b) (pc: P c)
(iac: immediate rel a c)
(ibc: immediate rel b c),
a = b.
Lemma path_ut :
∀ X (r r' : relation X) (T: transitive r') x y
(P: clos_refl_trans (fun x y ⇒ r x y ∨ r' x y) x y),
∃ z w,
clos_refl_trans r x z ∧
clos_refl_trans (fun x y ⇒ ∃ z, r' x z ∧ clos_trans r z y) z w ∧
(w = y ∨ r' w y).
Lemma path_ut2 :
∀ X (r r' : relation X) (T: transitive r') x y
(P: clos_trans (fun x y ⇒ r x y ∨ r' x y) x y),
clos_trans r x y ∨
∃ z w w',
clos_refl_trans r x z ∧
clos_refl_trans (fun x y ⇒ ∃ z, r' x z ∧ clos_trans r z y) z w ∧
r' w w' ∧
clos_refl_trans r w' y.
Lemma path_utd :
∀ X (r r' : relation X) (T: transitive r') dom
(F: is_total dom r')
(R: ∀ a b, r' a b → dom a ∧ dom b) x y
(P: clos_trans (fun x y ⇒ r x y ∨ r' x y) x y),
clos_trans r x y ∨
(∃ z w, clos_refl_trans r x z ∧ r' z w ∧ clos_refl_trans r w y) ∨
(∃ z w, r' z w ∧ clos_refl_trans r w z).
Lemma cycle_utd :
∀ X (r: relation X) (A: acyclic r)
r' (T: transitive r') (IRR: irreflexive r') dom
(F: is_total dom r')
(R: ∀ a b, r' a b → dom a ∧ dom b) x
(P: clos_trans (fun x y ⇒ r x y ∨ r' x y) x x),
∃ z w, r' z w ∧ clos_trans r w z.
Lemma acyclic_case_split A (R : relation A) f :
acyclic R ↔
acyclic (restr_rel f R) ∧ (∀ x (NEG: ¬ f x) (CYC: clos_trans R x x), False).
Lemma seqA2 X (r r' r'' : relation X) x y :
seq (seq r r') r'' x y ↔ seq r (seq r' r'') x y.
Lemma path_unc X (r r' : relation X)
(A: seq r r <--> (fun x y ⇒ False))
(B: seq r' r' <--> (fun x y ⇒ False)) :
clos_refl_trans (union r r') <-->
clos_refl_trans (seq r r') +++
(clos_refl_trans (seq r' r) +++
(seq r (clos_refl_trans (seq r' r)) +++
seq r' (clos_refl_trans (seq r r')))).
Lemma pathp_unc X (r r' : relation X)
(A: seq r r <--> (fun x y ⇒ False))
(B: seq r' r' <--> (fun x y ⇒ False)) :
clos_trans (union r r') <-->
clos_trans (seq r r') +++
(clos_trans (seq r' r) +++
(seq r (clos_refl_trans (seq r' r)) +++
seq r' (clos_refl_trans (seq r r')))).
Lemma acyclic_unc X (r r' : relation X)
(A: seq r r <--> (fun x y ⇒ False))
(B: seq r' r' <--> (fun x y ⇒ False)) :
acyclic (union r r') ↔ acyclic (seq r r').
Lemma in_split_perm A (x : A) l (IN: In x l) :
∃ l', Permutation l (x :: l').
Lemma in_concat_iff A (a: A) ll :
In a (concat ll) ↔ ∃ l, In a l ∧ In l ll.
Lemma in_concat A (a: A) l ll :
In a l →
In l ll →
In a (concat ll).
Add Parametric Morphism X : (@concat X) with
signature (@Permutation (list X)) ==> (@Permutation X)
as concat_more.
Lemma NoDup_concat_simpl A (a : A) l1 l2 ll
(ND: NoDup (concat ll))
(K: In l1 ll) (K' : In a l1)
(L: In l2 ll) (L' : In a l2) :
l1 = l2.
Lemma NoDup_concatD A (l: list A) ll :
NoDup (concat ll) → In l ll → NoDup l.
Lemma NoDup_eq_simpl A l1 (a : A) l1' l2 l2'
(ND : NoDup (l1 ++ a :: l1'))
(L : l1 ++ a :: l1' = l2 ++ a :: l2') :
l1 = l2 ∧ l1' = l2'.
Construct a total order from a list of elements
Definition total_order_from_list A (l: list A) x y :=
∃ l1 l2 l3, l = l1 ++ x :: l2 ++ y :: l3.
Lemma total_order_from_list_cons :
∀ A (a : A) l x y,
total_order_from_list (a :: l) x y ↔
a = x ∧ In y l ∨ total_order_from_list l x y.
Lemma total_order_from_list_app :
∀ A (l1 l2: list A) x y,
total_order_from_list (l1 ++ l2) x y ↔
In x l1 ∧ In y l2 ∨
total_order_from_list l1 x y ∨
total_order_from_list l2 x y.
Lemma total_order_from_list_insert :
∀ A (l1: list A) a l2 x y,
total_order_from_list (l1 ++ l2) x y →
total_order_from_list (l1 ++ a :: l2) x y.
Lemma total_order_from_list_remove :
∀ A (l1: list A) a l2 x y,
total_order_from_list (l1 ++ a :: l2) x y →
x ≠ a → y ≠ a →
total_order_from_list (l1 ++ l2) x y.
Lemma total_order_from_list_swap :
∀ A (l1: list A) a b l2 x y,
total_order_from_list (l1 ++ a :: b :: l2) x y →
(x = a → b = y → False) →
total_order_from_list (l1 ++ b :: a :: l2) x y.
Lemma total_order_from_list_in A (l: list A) x y :
total_order_from_list l x y → In x l ∧ In y l.
Lemma total_order_from_list_in1 A (l: list A) x y :
total_order_from_list l x y → In x l.
Lemma total_order_from_list_in2 A (l: list A) x y :
total_order_from_list l x y → In y l.
Lemma total_order_from_list_trans A (l : list A) (ND: NoDup l) x y z :
total_order_from_list l x y →
total_order_from_list l y z →
total_order_from_list l x z.
Lemma total_order_from_list_irreflexive A (l : list A) (ND: NoDup l) :
irreflexive (total_order_from_list l).
Lemma total_order_from_list_helper A (l : list A) (ND: NoDup l) :
∀ a b (IMM: immediate (total_order_from_list l) a b),
(∀ x, total_order_from_list l a x ↔ x = b ∨ total_order_from_list l b x) ∧
(∀ x, total_order_from_list l x b ↔ x = a ∨ total_order_from_list l x a).
Construct a union of total orders from a list of element lists
Definition mk_tou A (ll: list (list A)) x y :=
∃ l, In l ll ∧ total_order_from_list l x y.
Lemma mk_tou_trans A (ll : list (list A)) (ND: NoDup (concat ll)) x y z :
mk_tou ll x y →
mk_tou ll y z →
mk_tou ll x z.
Lemma mk_tou_irreflexive A (ll : list (list A)) (ND: NoDup (concat ll)) :
irreflexive (mk_tou ll).
Lemma mk_tou_in1 A ll (x y : A) :
mk_tou ll x y → In x (concat ll).
Lemma mk_tou_in2 A ll (x y : A) :
mk_tou ll x y → In y (concat ll).
Lemma mk_tou_trivial A ll1 l1 l2 ll2 (a b : A) :
mk_tou (ll1 ++ (l1 ++ a :: b :: l2) :: ll2) a b.
Lemma mk_tou_immediateD A ll (a b : A) :
immediate (mk_tou ll) a b →
∃ ll1 l1 l2 ll2, ll = ll1 ++ (l1 ++ a :: b :: l2) :: ll2.
Lemma mk_tou_immediate A ll1 l1 l2 ll2 (a b : A) :
NoDup (concat (ll1 ++ (l1 ++ a :: b :: l2) :: ll2)) →
immediate (mk_tou (ll1 ++ (l1 ++ a :: b :: l2) :: ll2)) a b.
Lemma mk_tou_helper A (ll : list (list A)) (ND: NoDup (concat ll)) :
∀ a b (IMM: immediate (mk_tou ll) a b),
(∀ x, mk_tou ll a x ↔ x = b ∨ mk_tou ll b x) ∧
(∀ x, mk_tou ll x b ↔ x = a ∨ mk_tou ll x a).
Lemma mk_tou_insert :
∀ A ll1 (l1: list A) a l2 ll2 x y,
mk_tou (ll1 ++ (l1 ++ l2) :: ll2) x y →
mk_tou (ll1 ++ (l1 ++ a :: l2) :: ll2) x y.
Lemma mk_tou_remove :
∀ A ll1 (l1: list A) a l2 ll2 x y,
mk_tou (ll1 ++ (l1 ++ a :: l2) :: ll2) x y →
x ≠ a → y ≠ a →
mk_tou (ll1 ++ (l1 ++ l2) :: ll2) x y.
Lemma mk_tou_swap :
∀ A ll1 (l1: list A) a b l2 ll2 x y,
mk_tou (ll1 ++ (l1 ++ a :: b :: l2) :: ll2) x y →
(x = a → b = y → False) →
mk_tou (ll1 ++ (l1 ++ b :: a :: l2) :: ll2) x y.
Definition mk_po A init ll (x y: A) :=
In x init ∧ In y (concat ll) ∨ mk_tou ll x y.
Lemma mk_po_trans A init ll (D: NoDup (init ++ concat ll)) (x y z : A) :
mk_po init ll x y →
mk_po init ll y z →
mk_po init ll x z.
Lemma transitive_mk_po A (i: list A) ll :
NoDup (i ++ concat ll) →
transitive (mk_po i ll).
Lemma mk_po_irreflexive A (init : list A) ll
(ND: NoDup (init ++ concat ll)) x :
mk_po init ll x x →
False.
Lemma mk_po_helper A init (ll : list (list A)) (ND: NoDup (init ++ concat ll)) :
∀ a (NI: ¬ In a init) b (IMM: immediate (mk_po init ll) a b),
(∀ x, mk_po init ll a x ↔ x = b ∨ mk_po init ll b x) ∧
(∀ x, mk_po init ll x b ↔ x = a ∨ mk_po init ll x a).
Lemma mk_po_in1 A init ll (x y : A) :
mk_po init ll x y → In x (init ++ concat ll).
Lemma mk_po_in2 A init ll (x y : A) :
mk_po init ll x y → In y (concat ll).
Lemma mk_po_in2_weak A init ll (x y : A) :
mk_po init ll x y → In y (init ++ concat ll).
Lemma mk_po_trivial A init ll1 l1 l2 ll2 (a b : A) :
mk_po init (ll1 ++ (l1 ++ a :: b :: l2) :: ll2) a b.
Lemma mk_po_immediateD A init ll (a b : A) :
immediate (mk_po init ll) a b →
¬ In a init →
∃ ll1 l1 l2 ll2, ll = ll1 ++ (l1 ++ a :: b :: l2) :: ll2.
Lemma mk_po_immediate A init ll1 l1 l2 ll2 (a b : A) :
NoDup (init ++ concat (ll1 ++ (l1 ++ a :: b :: l2) :: ll2)) →
immediate (mk_po init (ll1 ++ (l1 ++ a :: b :: l2) :: ll2)) a b.
Lemma mk_po_insert :
∀ A init ll1 (l1: list A) a l2 ll2 x y,
mk_po init (ll1 ++ (l1 ++ l2) :: ll2) x y →
mk_po init (ll1 ++ (l1 ++ a :: l2) :: ll2) x y.
Lemma mk_po_remove :
∀ A init ll1 (l1: list A) a l2 ll2 x y,
mk_po init (ll1 ++ (l1 ++ a :: l2) :: ll2) x y →
x ≠ a → y ≠ a →
mk_po init (ll1 ++ (l1 ++ l2) :: ll2) x y.
Lemma mk_po_swap :
∀ A init ll1 (l1: list A) a b l2 ll2 x y,
mk_po init (ll1 ++ (l1 ++ a :: b :: l2) :: ll2) x y →
(x = a → b = y → False) →
mk_po init (ll1 ++ (l1 ++ b :: a :: l2) :: ll2) x y.
Reordering of adjacent actions in a partial order
Section ReorderSection.
Variable A : Type.
Implicit Types po : relation A.
Implicit Types a b : A.
Definition reorder po a b x y :=
po x y ∧ ¬ (x = a ∧ y = b) ∨ x = b ∧ y = a.
Lemma reorderK po a b (NIN: ¬ po b a) (IN: po a b) :
reorder (reorder po a b) b a <--> po.
Lemma Permutation_reord i ll1 l1 a b l2 ll2 :
Permutation (i ++ concat (ll1 ++ (l1 ++ b :: a :: l2) :: ll2))
(i ++ concat (ll1 ++ (l1 ++ a :: b :: l2) :: ll2)).
Lemma mk_po_reorder init ll1 l1 a b l2 ll2 :
NoDup (init ++ concat (ll1 ++ (l1 ++ b :: a :: l2) :: ll2)) →
reorder (mk_po init (ll1 ++ (l1 ++ a :: b :: l2) :: ll2)) a b <-->
mk_po init (ll1 ++ (l1 ++ b :: a :: l2) :: ll2).
End ReorderSection.
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