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HITs-Examples
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Added lattice constructions

This commit is contained in:
Niels 2017-08-02 14:21:12 +02:00
parent 4141f9d456
commit 77a449e68b
1 changed files with 224 additions and 20 deletions
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244
FiniteSets/lattice.v
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@ -51,8 +51,8 @@ Section Lattice.
associative_max :> Associative max ;
idempotent_min :> Idempotent min ;
idempotent_max :> Idempotent max ;
neutralL_min :> NeutralL max empty ;
neutralR_min :> NeutralR max empty ;
neutralL_max :> NeutralL max empty ;
neutralR_max :> NeutralR max empty ;
absorption_min_max :> Absorption min max ;
absorption_max_min :> Absorption max min
}.
@ -64,7 +64,7 @@ Arguments Lattice {_} _ _ _.
Section BoolLattice.
Ltac solve :=
Ltac solve_bool :=
let x := fresh in
repeat (intro x ; destruct x)
; compute
@ -73,52 +73,52 @@ Section BoolLattice.
Instance orb_com : Commutative orb.
Proof.
solve.
solve_bool.
Defined.
Instance andb_com : Commutative andb.
Proof.
solve.
solve_bool.
Defined.
Instance orb_assoc : Associative orb.
Proof.
solve.
solve_bool.
Defined.
Instance andb_assoc : Associative andb.
Proof.
solve.
solve_bool.
Defined.
Instance orb_idem : Idempotent orb.
Proof.
solve.
solve_bool.
Defined.
Instance andb_idem : Idempotent andb.
Proof.
solve.
solve_bool.
Defined.
Instance orb_nl : NeutralL orb false.
Proof.
solve.
solve_bool.
Defined.
Instance orb_nr : NeutralR orb false.
Proof.
solve.
solve_bool.
Defined.
Instance bool_absorption_orb_andb : Absorption orb andb.
Proof.
solve.
solve_bool.
Defined.
Instance bool_absorption_andb_orb : Absorption andb orb.
Proof.
solve.
solve_bool.
Defined.
Global Instance lattice_bool : Lattice andb orb false :=
@ -128,42 +128,246 @@ Section BoolLattice.
associative_max := _ ;
idempotent_min := _ ;
idempotent_max := _ ;
neutralL_min := _ ;
neutralR_min := _ ;
neutralL_max := _ ;
neutralR_max := _ ;
absorption_min_max := _ ;
absorption_max_min := _
}.
Definition and_true : forall b, andb b true = b.
Proof.
solve.
solve_bool.
Defined.
Definition and_false : forall b, andb b false = false.
Proof.
solve.
solve_bool.
Defined.
Definition dist₁ : forall b₁ b₂ b₃,
andb b₁ (orb b₂ b₃) = orb (andb b₁ b₂) (andb b₁ b₃).
Proof.
solve.
solve_bool.
Defined.
Definition dist₂ : forall b₁ b₂ b₃,
orb b₁ (andb b₂ b₃) = andb (orb b₁ b₂) (orb b₁ b₃).
Proof.
solve.
solve_bool.
Defined.
Definition max_min : forall b₁ b₂,
orb (andb b₁ b₂) b₁ = b₁.
Proof.
solve.
solve_bool.
Defined.
End BoolLattice.
Section fun_lattice.
Context {A B : Type} {maxB minB : B -> B -> B} {botB : B}.
Context `{Lattice B minB maxB botB}.
Context `{Funext}.
Definition max_fun (f g : (A -> B)) (a : A) : B
:= maxB (f a) (g a).
Definition min_fun (f g : (A -> B)) (a : A) : B
:= minB (f a) (g a).
Definition bot_fun (a : A) : B
:= botB.
Hint Unfold max_fun min_fun bot_fun.
Ltac solve_fun := compute ; intros ; apply path_forall ; intro.
Instance max_fun_com : Commutative max_fun.
Proof.
solve_fun.
refine (commutative_max _ _ _ _ _ _).
Defined.
Instance min_fun_com : Commutative min_fun.
Proof.
solve_fun.
refine (commutative_min _ _ _ _ _ _).
Defined.
Instance max_fun_assoc : Associative max_fun.
Proof.
solve_fun.
refine (associative_max _ _ _ _ _ _ _).
Defined.
Instance min_fun_assoc : Associative min_fun.
Proof.
solve_fun.
refine (associative_min _ _ _ _ _ _ _).
Defined.
Instance max_fun_idem : Idempotent max_fun.
Proof.
solve_fun.
refine (idempotent_max _ _ _ _ _).
Defined.
Instance min_fun_idem : Idempotent min_fun.
Proof.
solve_fun.
refine (idempotent_min _ _ _ _ _).
Defined.
Instance max_fun_nl : NeutralL max_fun bot_fun.
Proof.
solve_fun.
simple refine (neutralL_max _ _ _ _ _).
Defined.
Instance max_fun_nr : NeutralR max_fun bot_fun.
Proof.
solve_fun.
simple refine (neutralR_max _ _ _ _ _).
Defined.
Instance absorption_max_fun_min_fun : Absorption max_fun min_fun.
Proof.
solve_fun.
simple refine (absorption_max_min _ _ _ _ _ _).
Defined.
Instance absorption_min_fun_max_fun : Absorption min_fun max_fun.
Proof.
solve_fun.
simple refine (absorption_min_max _ _ _ _ _ _).
Defined.
Global Instance lattice_fun : Lattice min_fun max_fun bot_fun :=
{ commutative_min := _ ;
commutative_max := _ ;
associative_min := _ ;
associative_max := _ ;
idempotent_min := _ ;
idempotent_max := _ ;
neutralL_max := _ ;
neutralR_max := _ ;
absorption_min_max := _ ;
absorption_max_min := _
}.
End fun_lattice.
Section sub_lattice.
Context {A : Type} {P : A -> hProp} {maxA minA : A -> A -> A} {botA : A}.
Context {Hmax : forall x y, P x -> P y -> P (maxA x y)}.
Context {Hmin : forall x y, P x -> P y -> P (minA x y)}.
Context {Hbot : P botA}.
Context `{Lattice A minA maxA botA}.
Definition AP : Type := sig P.
Definition botAP : AP := (botA ; Hbot).
Definition maxAP (x y : AP) : AP :=
match x with
| (a ; pa) => match y with
| (b ; pb) => (maxA a b ; Hmax a b pa pb)
end
end.
Definition minAP (x y : AP) : AP :=
match x with
| (a ; pa) => match y with
| (b ; pb) => (minA a b ; Hmin a b pa pb)
end
end.
Hint Unfold maxAP minAP botAP.
Instance hprop_sub : forall c, IsHProp (P c).
Proof.
apply _.
Defined.
Ltac solve_sub :=
let x := fresh in
repeat (intro x ; destruct x)
; simple refine (path_sigma _ _ _ _ _) ; try (apply hprop_sub).
Instance max_sub_com : Commutative maxAP.
Proof.
solve_sub.
refine (commutative_max _ _ _ _ _ _).
Defined.
Instance min_sub_com : Commutative minAP.
Proof.
solve_sub.
refine (commutative_min _ _ _ _ _ _).
Defined.
Instance max_sub_assoc : Associative maxAP.
Proof.
solve_sub.
refine (associative_max _ _ _ _ _ _ _).
Defined.
Instance min_sub_assoc : Associative minAP.
Proof.
solve_sub.
refine (associative_min _ _ _ _ _ _ _).
Defined.
Instance max_sub_idem : Idempotent maxAP.
Proof.
solve_sub.
refine (idempotent_max _ _ _ _ _).
Defined.
Instance min_sub_idem : Idempotent minAP.
Proof.
solve_sub.
refine (idempotent_min _ _ _ _ _).
Defined.
Instance max_sub_nl : NeutralL maxAP botAP.
Proof.
solve_sub.
simple refine (neutralL_max _ _ _ _ _).
Defined.
Instance max_sub_nr : NeutralR maxAP botAP.
Proof.
solve_sub.
simple refine (neutralR_max _ _ _ _ _).
Defined.
Instance absorption_max_sub_min_sub : Absorption maxAP minAP.
Proof.
solve_sub.
simple refine (absorption_max_min _ _ _ _ _ _).
Defined.
Instance absorption_min_sub_max_sub : Absorption minAP maxAP.
Proof.
solve_sub.
simple refine (absorption_min_max _ _ _ _ _ _).
Defined.
Global Instance lattice_sub : Lattice minAP maxAP botAP :=
{ commutative_min := _ ;
commutative_max := _ ;
associative_min := _ ;
associative_max := _ ;
idempotent_min := _ ;
idempotent_max := _ ;
neutralL_max := _ ;
neutralR_max := _ ;
absorption_min_max := _ ;
absorption_max_min := _
}.
End sub_lattice.
Hint Resolve
orb_com andb_com orb_assoc andb_assoc orb_idem andb_idem orb_nl orb_nr
bool_absorption_orb_andb bool_absorption_andb_orb and_true and_false