A negligible change in the structure

2025-12-15 22:53:51 +01:00 · 2017-09-07 15:19:48 +02:00
parent 4ab70ae1fe
commit 474c9324ca
29 changed files with 2082 additions and 1459 deletions
--- a/FiniteSets/interfaces/lattice_interface.v
+++ b/FiniteSets/interfaces/lattice_interface.v
@@ -0,0 +1,115 @@
+(** Interface for lattices and join semilattices. *)
+Require Import HoTT.
+
+Section binary_operation.
+  Definition operation (A : Type) := A -> A -> A.
+  
+  Variable (A : Type)
+           (f : operation A).
+
+  Class Commutative :=
+    commutative : forall x y, f x y = f y x.
+
+  Class Associative :=
+    associativity : forall x y z, f (f x y) z = f x (f y z).
+
+  Class Idempotent :=
+    idempotency : forall x, f x x = x.
+
+  Variable g : operation A.
+
+  Class Absorption :=
+    absorb : forall x y, f x (g x y) = x.
+
+  Variable (n : A).
+
+  Class NeutralL :=
+    neutralityL : forall x, f n x = x.
+
+  Class NeutralR :=
+    neutralityR : forall x, f x n = x.
+End binary_operation.
+
+Arguments Commutative {_} _.
+Arguments Associative {_} _.
+Arguments Idempotent {_} _.
+Arguments NeutralL {_} _ _.
+Arguments NeutralR {_} _ _.
+Arguments Absorption {_} _ _.
+Arguments commutative {_} {_} {_} _ _.
+Arguments associativity {_} {_} {_} _ _ _.
+Arguments idempotency {_} {_} {_} _.
+Arguments neutralityL {_} {_} {_} {_} _.
+Arguments neutralityR {_} {_} {_} {_} _.
+Arguments absorb {_} {_} {_} {_} _ _.
+
+Section lattice_operations.
+  Variable (A : Type).
+
+  Class maximum :=
+    max_L : operation A.
+
+  Class minimum :=
+    min_L : operation A.
+
+  Class bottom :=
+    empty : A.
+End lattice_operations.
+
+Arguments max_L {_} {_} _.
+Arguments min_L {_} {_} _.
+Arguments empty {_}.
+
+Section JoinSemiLattice.
+  Variable A : Type.
+  Context {max_L : maximum A} {empty_L : bottom A}.
+
+  Class JoinSemiLattice :=
+    {
+      commutative_max_js :> Commutative max_L ;
+      associative_max_js :> Associative max_L ;
+      idempotent_max_js :> Idempotent max_L ;
+      neutralL_max_js :> NeutralL max_L empty_L ;
+      neutralR_max_js :> NeutralR max_L empty_L ;
+    }.
+End JoinSemiLattice.
+
+Arguments JoinSemiLattice _ {_} {_}.
+
+Create HintDb joinsemilattice_hints.
+Hint Resolve associativity : joinsemilattice_hints.
+Hint Resolve (associativity _ _ _)^ : joinsemilattice_hints.
+Hint Resolve commutative : joinsemilattice_hints.
+Hint Resolve idempotency : joinsemilattice_hints.
+Hint Resolve neutralityL : joinsemilattice_hints.
+Hint Resolve neutralityR : joinsemilattice_hints.
+
+Section Lattice.
+  Variable A : Type.
+  Context {max_L : maximum A} {min_L : minimum A} {empty_L : bottom A}.
+
+  Class Lattice :=
+    {
+      commutative_min :> Commutative min_L ;
+      commutative_max :> Commutative max_L ;
+      associative_min :> Associative min_L ;
+      associative_max :> Associative max_L ;
+      idempotent_min :> Idempotent min_L ;
+      idempotent_max :> Idempotent max_L ;
+      neutralL_max :> NeutralL max_L empty_L ;
+      neutralR_max :> NeutralR max_L empty_L ;
+      absorption_min_max :> Absorption min_L max_L ;
+      absorption_max_min :> Absorption max_L min_L
+    }.
+End Lattice.
+
+Arguments Lattice _ {_} {_} {_}.
+
+Create HintDb lattice_hints.
+Hint Resolve associativity : lattice_hints.
+Hint Resolve (associativity _ _ _)^ : lattice_hints.
+Hint Resolve commutative : lattice_hints.
+Hint Resolve absorb : lattice_hints.
+Hint Resolve idempotency : lattice_hints.
+Hint Resolve neutralityL : lattice_hints.
+Hint Resolve neutralityR : lattice_hints.