mirror of https://github.com/nmvdw/HITs-Examples
376 lines
8.2 KiB
Coq
376 lines
8.2 KiB
Coq
(* Typeclass for lattices *)
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Require Import HoTT.
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Definition operation (A : Type) := A -> A -> A.
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Section Defs.
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Variable A : Type.
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Variable f : A -> A -> A.
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Class Commutative :=
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commutative : forall x y, f x y = f y x.
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Class Associative :=
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associativity : forall x y z, f (f x y) z = f x (f y z).
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Class Idempotent :=
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idempotency : forall x, f x x = x.
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Variable g : operation A.
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Class Absorption :=
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absrob : forall x y, f x (g x y) = x.
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Variable n : A.
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Class NeutralL :=
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neutralityL : forall x, f n x = x.
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Class NeutralR :=
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neutralityR : forall x, f x n = x.
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End Defs.
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Arguments Commutative {_} _.
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Arguments Associative {_} _.
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Arguments Idempotent {_} _.
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Arguments NeutralL {_} _ _.
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Arguments NeutralR {_} _ _.
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Arguments Absorption {_} _ _.
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Section Lattice.
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Variable A : Type.
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Variable min max : operation A.
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Variable empty : A.
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Class Lattice :=
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{
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commutative_min :> Commutative min ;
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commutative_max :> Commutative max ;
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associative_min :> Associative min ;
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associative_max :> Associative max ;
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idempotent_min :> Idempotent min ;
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idempotent_max :> Idempotent max ;
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neutralL_max :> NeutralL max empty ;
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neutralR_max :> NeutralR max empty ;
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absorption_min_max :> Absorption min max ;
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absorption_max_min :> Absorption max min
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}.
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End Lattice.
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Arguments Lattice {_} _ _ _.
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Section BoolLattice.
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Ltac solve_bool :=
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let x := fresh in
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repeat (intro x ; destruct x)
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; compute
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; auto
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; try contradiction.
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Instance orb_com : Commutative orb.
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Proof.
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solve_bool.
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Defined.
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Instance andb_com : Commutative andb.
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Proof.
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solve_bool.
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Defined.
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Instance orb_assoc : Associative orb.
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Proof.
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solve_bool.
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Defined.
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Instance andb_assoc : Associative andb.
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Proof.
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solve_bool.
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Defined.
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Instance orb_idem : Idempotent orb.
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Proof.
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solve_bool.
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Defined.
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Instance andb_idem : Idempotent andb.
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Proof.
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solve_bool.
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Defined.
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Instance orb_nl : NeutralL orb false.
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Proof.
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solve_bool.
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Defined.
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Instance orb_nr : NeutralR orb false.
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Proof.
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solve_bool.
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Defined.
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Instance bool_absorption_orb_andb : Absorption orb andb.
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Proof.
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solve_bool.
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Defined.
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Instance bool_absorption_andb_orb : Absorption andb orb.
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Proof.
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solve_bool.
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Defined.
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Global Instance lattice_bool : Lattice andb orb false :=
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{ commutative_min := _ ;
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commutative_max := _ ;
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associative_min := _ ;
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associative_max := _ ;
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idempotent_min := _ ;
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idempotent_max := _ ;
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neutralL_max := _ ;
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neutralR_max := _ ;
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absorption_min_max := _ ;
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absorption_max_min := _
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}.
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Definition and_true : forall b, andb b true = b.
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Proof.
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solve_bool.
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Defined.
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Definition and_false : forall b, andb b false = false.
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Proof.
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solve_bool.
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Defined.
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Definition dist₁ : forall b₁ b₂ b₃,
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andb b₁ (orb b₂ b₃) = orb (andb b₁ b₂) (andb b₁ b₃).
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Proof.
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solve_bool.
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Defined.
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Definition dist₂ : forall b₁ b₂ b₃,
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orb b₁ (andb b₂ b₃) = andb (orb b₁ b₂) (orb b₁ b₃).
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Proof.
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solve_bool.
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Defined.
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Definition max_min : forall b₁ b₂,
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orb (andb b₁ b₂) b₁ = b₁.
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Proof.
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solve_bool.
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Defined.
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End BoolLattice.
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Section fun_lattice.
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Context {A B : Type} {maxB minB : B -> B -> B} {botB : B}.
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Context `{Lattice B minB maxB botB}.
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Context `{Funext}.
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Definition max_fun (f g : (A -> B)) (a : A) : B
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:= maxB (f a) (g a).
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Definition min_fun (f g : (A -> B)) (a : A) : B
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:= minB (f a) (g a).
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Definition bot_fun (a : A) : B
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:= botB.
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Hint Unfold max_fun min_fun bot_fun.
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Ltac solve_fun := compute ; intros ; apply path_forall ; intro.
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Instance max_fun_com : Commutative max_fun.
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Proof.
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solve_fun.
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refine (commutative_max _ _ _ _ _ _).
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Defined.
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Instance min_fun_com : Commutative min_fun.
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Proof.
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solve_fun.
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refine (commutative_min _ _ _ _ _ _).
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Defined.
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Instance max_fun_assoc : Associative max_fun.
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Proof.
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solve_fun.
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refine (associative_max _ _ _ _ _ _ _).
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Defined.
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Instance min_fun_assoc : Associative min_fun.
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Proof.
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solve_fun.
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refine (associative_min _ _ _ _ _ _ _).
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Defined.
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Instance max_fun_idem : Idempotent max_fun.
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Proof.
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solve_fun.
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refine (idempotent_max _ _ _ _ _).
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Defined.
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Instance min_fun_idem : Idempotent min_fun.
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Proof.
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solve_fun.
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refine (idempotent_min _ _ _ _ _).
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Defined.
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Instance max_fun_nl : NeutralL max_fun bot_fun.
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Proof.
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solve_fun.
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simple refine (neutralL_max _ _ _ _ _).
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Defined.
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Instance max_fun_nr : NeutralR max_fun bot_fun.
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Proof.
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solve_fun.
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simple refine (neutralR_max _ _ _ _ _).
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Defined.
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Instance absorption_max_fun_min_fun : Absorption max_fun min_fun.
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Proof.
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solve_fun.
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simple refine (absorption_max_min _ _ _ _ _ _).
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Defined.
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Instance absorption_min_fun_max_fun : Absorption min_fun max_fun.
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Proof.
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solve_fun.
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simple refine (absorption_min_max _ _ _ _ _ _).
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Defined.
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Global Instance lattice_fun : Lattice min_fun max_fun bot_fun :=
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{ commutative_min := _ ;
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commutative_max := _ ;
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associative_min := _ ;
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associative_max := _ ;
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idempotent_min := _ ;
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idempotent_max := _ ;
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neutralL_max := _ ;
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neutralR_max := _ ;
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absorption_min_max := _ ;
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absorption_max_min := _
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}.
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End fun_lattice.
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Section sub_lattice.
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Context {A : Type} {P : A -> hProp} {maxA minA : A -> A -> A} {botA : A}.
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Context {Hmax : forall x y, P x -> P y -> P (maxA x y)}.
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Context {Hmin : forall x y, P x -> P y -> P (minA x y)}.
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Context {Hbot : P botA}.
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Context `{Lattice A minA maxA botA}.
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Definition AP : Type := sig P.
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Definition botAP : AP := (botA ; Hbot).
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Definition maxAP (x y : AP) : AP :=
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match x with
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| (a ; pa) => match y with
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| (b ; pb) => (maxA a b ; Hmax a b pa pb)
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end
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end.
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Definition minAP (x y : AP) : AP :=
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match x with
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| (a ; pa) => match y with
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| (b ; pb) => (minA a b ; Hmin a b pa pb)
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end
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end.
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Hint Unfold maxAP minAP botAP.
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Instance hprop_sub : forall c, IsHProp (P c).
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Proof.
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apply _.
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Defined.
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Ltac solve_sub :=
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let x := fresh in
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repeat (intro x ; destruct x)
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; simple refine (path_sigma _ _ _ _ _) ; try (apply hprop_sub).
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Instance max_sub_com : Commutative maxAP.
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Proof.
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solve_sub.
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refine (commutative_max _ _ _ _ _ _).
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Defined.
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Instance min_sub_com : Commutative minAP.
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Proof.
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solve_sub.
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refine (commutative_min _ _ _ _ _ _).
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Defined.
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Instance max_sub_assoc : Associative maxAP.
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Proof.
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solve_sub.
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refine (associative_max _ _ _ _ _ _ _).
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Defined.
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Instance min_sub_assoc : Associative minAP.
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Proof.
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solve_sub.
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refine (associative_min _ _ _ _ _ _ _).
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Defined.
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Instance max_sub_idem : Idempotent maxAP.
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Proof.
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solve_sub.
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refine (idempotent_max _ _ _ _ _).
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Defined.
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Instance min_sub_idem : Idempotent minAP.
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Proof.
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solve_sub.
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refine (idempotent_min _ _ _ _ _).
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Defined.
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Instance max_sub_nl : NeutralL maxAP botAP.
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Proof.
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solve_sub.
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simple refine (neutralL_max _ _ _ _ _).
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Defined.
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Instance max_sub_nr : NeutralR maxAP botAP.
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Proof.
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solve_sub.
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simple refine (neutralR_max _ _ _ _ _).
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Defined.
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Instance absorption_max_sub_min_sub : Absorption maxAP minAP.
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Proof.
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solve_sub.
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simple refine (absorption_max_min _ _ _ _ _ _).
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Defined.
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Instance absorption_min_sub_max_sub : Absorption minAP maxAP.
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Proof.
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solve_sub.
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simple refine (absorption_min_max _ _ _ _ _ _).
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Defined.
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Global Instance lattice_sub : Lattice minAP maxAP botAP :=
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{ commutative_min := _ ;
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commutative_max := _ ;
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associative_min := _ ;
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associative_max := _ ;
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idempotent_min := _ ;
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idempotent_max := _ ;
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neutralL_max := _ ;
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neutralR_max := _ ;
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absorption_min_max := _ ;
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absorption_max_min := _
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}.
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End sub_lattice.
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Hint Resolve
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orb_com andb_com orb_assoc andb_assoc orb_idem andb_idem orb_nl orb_nr
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bool_absorption_orb_andb bool_absorption_andb_orb and_true and_false
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dist₁ dist₂ max_min
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: bool_lattice_hints.
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