# Base Library for ICL

• Version: 27 April 2015
• Author: Gert Smolka, Saarland University
• Acknowlegments: Sigurd Schneider, Dominik Kirst

(* Switch Coq into implicit argument mode *)

Global Set Implicit Arguments.

(* Load basic Coq libraries *)

Require Export Omega List Morphisms.

(* Inversion tactic *)

Ltac inv H := inversion H; try subst; clear H.

# De Morgan laws

Lemma DM_or (X Y : Prop) :
¬ (X Y) ¬ X ¬ Y.

Lemma DM_exists X (p : X Prop) :
¬ ( x, p x) x, ¬ p x.

# Size recursion

Lemma size_recursion (X : Type) (sigma : X nat) (p : X Type) :
( x, ( y, sigma y < sigma x p y) p x)
x, p x.

# Iteration

Section Iteration.
Variable X : Type.
Variable f : X X.

Fixpoint it (n : nat) (x : X) : X :=
match n with
| 0 ⇒ x
| S n'f (it n' x)
end.

Lemma it_ind (p : X Prop) x n :
p x ( z, p z p (f z)) p (it n x).

Definition FP (x : X) : Prop := f x = x.

Lemma it_fp (sigma : X nat) x :
( n, FP (it n x) sigma (it n x) > sigma (it (S n) x))
FP (it (sigma x) x).
End Iteration.

# Decidability

Definition dec (X : Prop) : Type := {X} + {¬ X}.

Notation "'eq_dec' X" := ( x y : X, dec (x=y)) (at level 70).

(* Register dec as a type class *)

Definition decision (X : Prop) (D : dec X) : dec X := D.

Tactic Notation "decide" constr(p) :=
destruct (decision p).
Tactic Notation "decide" constr(p) "as" simple_intropattern(i) :=
destruct (decision p) as i.

(* Hints for auto concerning dec *)

Hint Extern 4 ⇒
match goal with
| [ |- dec ?p ] ⇒ exact (decision p)
end.

(* Improves type class inference *)

Hint Extern 4 ⇒
match goal with
| [ |- dec ((fun __) _) ] ⇒ simpl
end : typeclass_instances.

(* Register instance rules for dec *)

Instance True_dec : dec True :=
left I.

Instance False_dec : dec False :=
right (fun AA).

Instance impl_dec (X Y : Prop) :
dec X dec Y dec (X Y).

Instance and_dec (X Y : Prop) :
dec X dec Y dec (X Y).

Instance or_dec (X Y : Prop) :
dec X dec Y dec (X Y).

(* Coq standard modules make "not" and "iff" opaque for type class inference, can be seen with Print HintDb typeclass_instances. *)

Instance not_dec (X : Prop) :
dec X dec (¬ X).

Instance iff_dec (X Y : Prop) :
dec X dec Y dec (X Y).

Lemma dec_DN X :
dec X ~~ X X.

Lemma dec_DM_and X Y :
dec X dec Y ¬ (X Y) ¬ X ¬ Y.

Lemma dec_DM_impl X Y :
dec X dec Y ¬ (X Y) X ¬ Y.

Lemma dec_prop_iff (X Y : Prop) :
(X Y) dec X dec Y.

Instance bool_eq_dec :
eq_dec bool.

Instance nat_eq_dec :
eq_dec nat.

Instance nat_le_dec (x y : nat) : dec (x y) :=
le_dec x y.

# Lists

Definition equi X (A B : list X) : Prop :=
incl A B incl B A.

Hint Unfold equi.

Export ListNotations.
Notation "| A |" := (length A) (at level 65).
Notation "x 'el' A" := (In x A) (at level 70).
Notation "A <<= B" := (incl A B) (at level 70).
Notation "A === B" := (equi A B) (at level 70).

(* The following comments are for coqdoc *)

Register additional simplification rules with autorewrite / simpl_list

Hint Rewrite <- app_assoc : list.
Hint Rewrite rev_app_distr map_app prod_length : list.
(* Print Rewrite HintDb list. *)

Lemma list_cycle (X : Type) (A : list X) x :
x::A A.

# Decidability laws for lists

Instance list_eq_dec X :
eq_dec X eq_dec (list X).

Instance list_in_dec (X : Type) (x : X) (A : list X) :
eq_dec X dec (x A).

Lemma list_sigma_forall X A (p : X Prop) (p_dec : x, dec (p x)) :
{x | x A p x} + { x, x A ¬ p x}.

Instance list_forall_dec X A (p : X Prop) :
( x, dec (p x)) dec ( x, x A p x).

Instance list_exists_dec X A (p : X Prop) :
( x, dec (p x)) dec ( x, x A p x).

Lemma list_exists_DM X A (p : X Prop) :
( x, dec (p x))
¬ ( x, x A ¬ p x) x, x A p x.

Lemma list_exists_not_incl X (A B : list X) :
eq_dec X
¬ A B x, x A ¬ x B.

Lemma list_cc X (p : X Prop) A :
( x, dec (p x))
( x, x A p x) {x | x A p x}.

# Membership

We use the following lemmas from Coq's standard library List.
• in_eq : x x::A
• in_nil : ¬ x nil
• in_cons : x A x y::A
• in_or_app : x A x B x A++B
• in_app_iff : x A++B x A x B
• in_map_iff : y map f A x, f x = y x A

Hint Resolve in_eq in_nil in_cons in_or_app.

Section Membership.
Variable X : Type.
Implicit Types x y : X.
Implicit Types A B : list X.

Lemma in_sing x y :
x [y] x = y.

Lemma in_cons_neq x y A :
x y::A x y x A.

Lemma not_in_cons x y A :
¬ x y :: A x y ¬ x A.

# Disjointness

Definition disjoint A B :=
¬ x, x A x B.

Lemma disjoint_forall A B :
disjoint A B x, x A ¬ x B.

Lemma disjoint_symm A B :
disjoint A B disjoint B A.

Lemma disjoint_incl A B B' :
B' B disjoint A B disjoint A B'.

Lemma disjoint_nil B :
disjoint nil B.

Lemma disjoint_nil' A :
disjoint A nil.

Lemma disjoint_cons x A B :
disjoint (x::A) B ¬ x B disjoint A B.

Lemma disjoint_app A B C :
disjoint (A ++ B) C disjoint A C disjoint B C.

End Membership.

Hint Resolve disjoint_nil disjoint_nil'.

# Inclusion

We use the following lemmas from Coq's standard library List.
• incl_refl : A A
• incl_tl : A B A x::B
• incl_cons : x B A B x::A B
• incl_appl : A B A B++C
• incl_appr : A C A B++C
• incl_app : A C B C A++B C

Hint Resolve incl_refl incl_tl incl_cons incl_appl incl_appr incl_app.

Lemma incl_nil X (A : list X) :
nil A.

Hint Resolve incl_nil.

Lemma incl_map X Y A B (f : X Y) :
A B map f A map f B.

Section Inclusion.
Variable X : Type.
Implicit Types A B : list X.

Lemma incl_nil_eq A :
A nil A=nil.

Lemma incl_shift x A B :
A B x::A x::B.

Lemma incl_lcons x A B :
x::A B x B A B.

Lemma incl_sing x A y :
x::A [y] x = y A [y].

Lemma incl_rcons x A B :
A x::B ¬ x A A B.

Lemma incl_lrcons x A B :
x::A x::B ¬ x A A B.

Lemma incl_app_left A B C :
A ++ B C A C B C.

End Inclusion.

Definition inclp (X : Type) (A : list X) (p : X Prop) : Prop :=
x, x A p x.

# Setoid rewriting with list inclusion and list equivalence

Instance incl_preorder X :
PreOrder (@incl X).

Instance equi_Equivalence X :
Equivalence (@equi X).

Instance incl_equi_proper X :
Proper (@equi X ==> @equi X ==> iff) (@incl X).

Instance cons_incl_proper X x :
Proper (@incl X ==> @incl X) (@cons X x).

Instance cons_equi_proper X x :
Proper (@equi X ==> @equi X) (@cons X x).

Instance in_incl_proper X x :
Proper (@incl X ==> Basics.impl) (@In X x).

Instance in_equi_proper X x :
Proper (@equi X ==> iff) (@In X x).

Instance app_incl_proper X :
Proper (@incl X ==> @incl X ==> @incl X) (@app X).

Instance app_equi_proper X :
Proper (@equi X ==> @equi X ==> @equi X) (@app X).

# Equivalence

Section Equi.
Variable X : Type.
Implicit Types A B : list X.

Lemma equi_push x A :
x A A x::A.

Lemma equi_dup x A :
x::A x::x::A.

Lemma equi_swap x y A:
x::y::A y::x::A.

Lemma equi_shift x A B :
x::A++B A++x::B.

Lemma equi_rotate x A :
x::A A++[x].
End Equi.

# Filter

Definition filter (X : Type) (p : X Prop) (p_dec : x, dec (p x)) : list X list X :=
fix f A := match A with
| nilnil
| x::A'if decision (p x) then x :: f A' else f A'
end.

Section FilterLemmas.
Variable X : Type.
Variable p : X Prop.
Context {p_dec : x, dec (p x)}.

Lemma in_filter_iff x A :
x filter p A x A p x.

Lemma filter_incl A :
filter p A A.

Lemma filter_mono A B :
A B filter p A filter p B.

Lemma filter_id A :
( x, x A p x) filter p A = A.

Lemma filter_app A B :
filter p (A ++ B) = filter p A ++ filter p B.

Lemma filter_fst x A :
p x filter p (x::A) = x::filter p A.

Lemma filter_fst' x A :
¬ p x filter p (x::A) = filter p A.

End FilterLemmas.

Section FilterLemmas_pq.
Variable X : Type.
Variable p q : X Prop.
Context {p_dec : x, dec (p x)}.
Context {q_dec : x, dec (q x)}.

Lemma filter_pq_mono A :
( x, x A p x q x) filter p A filter q A.

Lemma filter_pq_eq A :
( x, x A (p x q x)) filter p A = filter q A.

Lemma filter_and A :
filter p (filter q A) = filter (fun xp x q x) A.

End FilterLemmas_pq.

Section FilterComm.
Variable X : Type.
Variable p q : X Prop.
Context {p_dec : x, dec (p x)}.
Context {q_dec : x, dec (q x)}.

Lemma filter_comm A :
filter p (filter q A) = filter q (filter p A).
End FilterComm.

# Element removal

Section Removal.
Variable X : Type.
Context {eq_X_dec : eq_dec X}.

Definition rem (A : list X) (x : X) : list X :=
filter (fun zz x) A.

Lemma in_rem_iff x A y :
x rem A y x A x y.

Lemma rem_not_in x y A :
x = y ¬ x A ¬ x rem A y.

Lemma rem_incl A x :
rem A x A.

Lemma rem_mono A B x :
A B rem A x rem B x.

Lemma rem_cons A B x :
A B rem (x::A) x B.

Lemma rem_cons' A B x y :
x B rem A y B rem (x::A) y B.

Lemma rem_in x y A :
x rem A y x A.

Lemma rem_neq x y A :
x y x A x rem A y.

Lemma rem_app x A B :
x A B A ++ rem B x.

Lemma rem_app' x A B C :
rem A x C rem B x C rem (A ++ B) x C.

Lemma rem_equi x A :
x::A x::rem A x.

Lemma rem_comm A x y :
rem (rem A x) y = rem (rem A y) x.

Lemma rem_fst x A :
rem (x::A) x = rem A x.

Lemma rem_fst' x y A :
x y rem (x::A) y = x::rem A y.

Lemma rem_id x A :
¬ x A rem A x = A.

Lemma rem_reorder x A :
x A A x :: rem A x.

Lemma rem_inclr A B x :
A B ¬ x A A rem B x.

End Removal.

Hint Resolve rem_not_in rem_incl rem_mono rem_cons rem_cons' rem_app rem_app' rem_in rem_neq rem_inclr.

# Cardinality

Section Cardinality.
Variable X : Type.
Context { eq_X_dec : eq_dec X }.
Implicit Types A B : list X.

Fixpoint card A :=
match A with
| nil ⇒ 0
| x::Aif decision (x A) then card A else 1 + card A
end.

Lemma card_in_rem x A :
x A card A = 1 + card (rem A x).

Lemma card_not_in_rem A x :
¬ x A card A = card (rem A x).

Lemma card_le A B :
A B card A card B.

Lemma card_eq A B :
A B card A = card B.

Lemma card_cons_rem x A :
card (x::A) = 1 + card (rem A x).

Lemma card_0 A :
card A = 0 A = nil.

Lemma card_ex A B :
card A < card B x, x B ¬ x A.

Lemma card_equi A B :
A B card A = card B A B.

Lemma card_lt A B x :
A B x B ¬ x A card A < card B.

Lemma card_or A B :
A B A B card A < card B.

End Cardinality.

Instance card_equi_proper X (D: eq_dec X) :
Proper (@equi X ==> eq) (@card X D).

# Duplicate-free lists

Inductive dupfree (X : Type) : list X Prop :=
| dupfreeN : dupfree nil
| dupfreeC x A : ¬ x A dupfree A dupfree (x::A).

Section Dupfree.
Variable X : Type.
Implicit Types A B : list X.

Lemma dupfree_cons x A :
dupfree (x::A) ¬ x A dupfree A.

Lemma dupfree_app A B :
disjoint A B dupfree A dupfree B dupfree (A++B).

Lemma dupfree_map Y A (f : X Y) :
( x y, x A y A f x = f y x=y)
dupfree A dupfree (map f A).

Lemma dupfree_filter p (p_dec : x, dec (p x)) A :
dupfree A dupfree (filter p A).

Lemma dupfree_dec A :
eq_dec X dec (dupfree A).

Lemma dupfree_card A (eq_X_dec : eq_dec X) :
dupfree A card A = |A|.

End Dupfree.

Section Undup.
Variable X : Type.
Context {eq_X_dec : eq_dec X}.
Implicit Types A B : list X.

Fixpoint undup (A : list X) : list X :=
match A with
| nilnil
| x::A'if decision (x A') then undup A' else x :: undup A'
end.

Lemma undup_id_equi A :
undup A A.

Lemma dupfree_undup A :
dupfree (undup A).

Lemma undup_incl A B :
A B undup A undup B.

Lemma undup_equi A B :
A B undup A undup B.

Lemma undup_id A :
dupfree A undup A = A.

Lemma undup_idempotent A :
undup (undup A) = undup A.

End Undup.

# Power lists

Section PowerRep.
Variable X : Type.
Context {eq_X_dec : eq_dec X}.

Fixpoint power (U : list X ) : list (list X) :=
match U with
| nil[nil]
| x :: U'power U' ++ map (cons x) (power U')
end.

Lemma power_incl A U :
A power U A U.

Lemma power_nil U :
nil power U.

Definition rep (A U : list X) : list X :=
filter (fun xx A) U.

Lemma rep_power A U :
rep A U power U.

Lemma rep_incl A U :
rep A U A.

Lemma rep_in x A U :
A U x A x rep A U.

Lemma rep_equi A U :
A U rep A U A.

Lemma rep_mono A B U :
A B rep A U rep B U.

Lemma rep_eq' A B U :
( x, x U (x A x B)) rep A U = rep B U.

Lemma rep_eq A B U :
A B rep A U = rep B U.

Lemma rep_injective A B U :
A U B U rep A U = rep B U A B.

Lemma rep_idempotent A U :
rep (rep A U) U = rep A U.

Lemma dupfree_power U :
dupfree U dupfree (power U).

Lemma dupfree_in_power U A :
A power U dupfree U dupfree A.

Lemma rep_dupfree A U :
dupfree U A power U rep A U = A.

Lemma power_extensional A B U :
dupfree U A power U B power U A B A = B.

End PowerRep.

# Finite closure iteration

Module FCI.
Section FCI.
Variable X : Type.
Context {eq_X_dec : eq_dec X}.
Variable step : list X X Prop.
Context {step_dec : A x, dec (step A x)}.
Variable V : list X.

Lemma pick (A : list X) :
{ x | x V step A x ¬ x A } + { x, x V step A x x A }.

Definition F (A : list X) : list X.
Defined.

Definition C := it F (card V) nil.

Lemma it_incl n :
it F n nil V.

Lemma incl :
C V.

Lemma ind p :
( A x, inclp A p x V step A x p x) inclp C p.

Lemma fp :
F C = C.

Lemma closure x :
x V step C x x C.

End FCI.
End FCI.

# Deprecated names, defined for backward compatibilitly

Definition dupfree_inv := dupfree_cons.