mirror of https://github.com/nmvdw/HITs-Examples
187 lines
4.3 KiB
Coq
187 lines
4.3 KiB
Coq
(* Logical disjunction in HoTT (see ch. 3 of the book) *)
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Require Import HoTT.
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Require Import lattice notation.
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Instance lor : maximum hProp := fun X Y => BuildhProp (Trunc (-1) (sum X Y)).
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Delimit Scope logic_scope with L.
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Notation "A ∨ B" := (lor A B) (at level 20, right associativity) : logic_scope.
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Arguments lor _%L _%L.
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Open Scope logic_scope.
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Section lor_props.
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Context `{Univalence}.
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Variable X Y Z : hProp.
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Lemma lor_assoc : (X ∨ Y) ∨ Z = X ∨ Y ∨ Z.
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Proof.
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apply path_iff_hprop ; cbn.
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* simple refine (Trunc_ind _ _).
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intros [xy | z] ; cbn.
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+ simple refine (Trunc_ind _ _ xy).
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intros [x | y].
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++ apply (tr (inl x)).
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++ apply (tr (inr (tr (inl y)))).
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+ apply (tr (inr (tr (inr z)))).
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* simple refine (Trunc_ind _ _).
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intros [x | yz] ; cbn.
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+ apply (tr (inl (tr (inl x)))).
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+ simple refine (Trunc_ind _ _ yz).
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intros [y | z].
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++ apply (tr (inl (tr (inr y)))).
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++ apply (tr (inr z)).
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Defined.
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Lemma lor_comm : X ∨ Y = Y ∨ X.
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Proof.
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apply path_iff_hprop ; cbn.
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* simple refine (Trunc_ind _ _).
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intros [x | y].
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+ apply (tr (inr x)).
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+ apply (tr (inl y)).
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* simple refine (Trunc_ind _ _).
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intros [y | x].
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+ apply (tr (inr y)).
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+ apply (tr (inl x)).
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Defined.
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Lemma lor_nl : False_hp ∨ X = X.
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Proof.
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apply path_iff_hprop ; cbn.
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* simple refine (Trunc_ind _ _).
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intros [ | x].
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+ apply Empty_rec.
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+ apply x.
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* apply (fun x => tr (inr x)).
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Defined.
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Lemma lor_nr : X ∨ False_hp = X.
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Proof.
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apply path_iff_hprop ; cbn.
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* simple refine (Trunc_ind _ _).
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intros [x | ].
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+ apply x.
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+ apply Empty_rec.
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* apply (fun x => tr (inl x)).
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Defined.
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Lemma lor_idem : X ∨ X = X.
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Proof.
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apply path_iff_hprop ; cbn.
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- simple refine (Trunc_ind _ _).
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intros [x | x] ; apply x.
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- apply (fun x => tr (inl x)).
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Defined.
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End lor_props.
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Instance land : minimum hProp := fun X Y => BuildhProp (prod X Y).
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Instance lfalse : bottom hProp := False_hp.
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Notation "A ∧ B" := (land A B) (at level 20, right associativity) : logic_scope.
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Arguments land _%L _%L.
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Open Scope logic_scope.
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Section hPropLattice.
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Context `{Univalence}.
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Instance lor_commutative : Commutative lor.
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Proof.
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unfold Commutative.
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intros.
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apply lor_comm.
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Defined.
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Instance land_commutative : Commutative land.
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Proof.
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unfold Commutative, land.
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intros.
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apply path_hprop.
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apply equiv_prod_symm.
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Defined.
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Instance lor_associative : Associative lor.
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Proof.
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unfold Associative.
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intros.
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apply lor_assoc.
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Defined.
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Instance land_associative : Associative land.
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Proof.
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unfold Associative, land.
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intros.
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symmetry.
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apply path_hprop.
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apply equiv_prod_assoc.
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Defined.
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Instance lor_idempotent : Idempotent lor.
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Proof.
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unfold Idempotent.
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intros.
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apply lor_idem.
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Defined.
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Instance land_idempotent : Idempotent land.
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Proof.
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unfold Idempotent, land.
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intros.
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apply path_iff_hprop ; cbn.
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- intros [a b] ; apply a.
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- intros a ; apply (pair a a).
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Defined.
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Instance lor_neutrall : NeutralL lor lfalse.
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Proof.
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unfold NeutralL.
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intros.
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apply lor_nl.
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Defined.
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Instance lor_neutralr : NeutralR lor lfalse.
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Proof.
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unfold NeutralR.
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intros.
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apply lor_nr.
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Defined.
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Instance absorption_orb_andb : Absorption lor land.
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Proof.
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unfold Absorption.
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intros.
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apply path_iff_hprop ; cbn.
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- intros X ; strip_truncations.
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destruct X as [Xx | [Xy1 Xy2]] ; assumption.
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- intros X.
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apply (tr (inl X)).
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Defined.
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Instance absorption_andb_orb : Absorption land lor.
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Proof.
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unfold Absorption.
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intros.
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apply path_iff_hprop ; cbn.
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- intros [X Y] ; strip_truncations.
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assumption.
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- intros X.
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split.
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* assumption.
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* apply (tr (inl X)).
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Defined.
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Global Instance lattice_hprop : Lattice hProp :=
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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 hPropLattice.
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