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basics.v
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basics.v
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Require Import HoTT FunextAxiom UnivalenceAxiom.
Require Import unique_choice.
Require Export structures.
Instance neq_is_prop : forall {T}, Relation.RelationProp (@neq T).
Proof.
red;intros. apply trunc_forall.
Defined.
Instance iff_trans : Transitive iff.
Proof.
red. intros A B C [H H'] [H0 H0'].
split;auto.
Defined.
Instance iff_symm : Symmetric iff.
Proof.
red. intros A B [H H'];split;assumption.
Defined.
Instance iff_refl : Reflexive iff.
Proof.
intros A. split;exact idmap.
Defined.
Module Magma_pr.
Export Magma.
Instance law_set : forall A, IsHSet A -> IsHSet (law A).
Proof.
intros.
apply @trunc_arrow. exact _.
apply trunc_arrow.
Defined.
Instance assoc_prop : forall {G} {L : Gop G} {Hset : IsHSet G},
IsHProp (Associative L).
Proof.
intros.
apply trunc_forall.
Defined.
Instance comm_prop : forall {G} {L : Gop G} {Hset : IsHSet G},
IsHProp (Commutative L).
Proof.
intros.
apply trunc_forall.
Defined.
Instance issg_prop : forall {G} {L : Gop G} {Hset : IsHSet G},
IsHProp (IsSemigroup L).
Proof.
intros G L Hset.
apply (@trunc_equiv' (sigT (fun _ : Associative L => Commutative L))).
issig (BuildIsSemigroup L) (@sg_assoc G L) (@sg_comm G L).
apply @trunc_sigma. apply _. intro. apply _.
Defined.
Lemma id_unique : forall {G} (L : Gop G), forall e : G,
IsId L e -> forall e': G, IsId L e' -> e = e'.
Proof.
intros.
apply (@concat _ _ (gop e e')).
apply inverse;apply right_id. apply _.
apply left_id. apply _.
Defined.
Instance id_prop : forall {G} {L : Gop G} {Hset : IsHSet G} e,
IsHProp (IsId L e).
Proof.
intros.
apply (@trunc_equiv' (sigT (fun _ : Left_id L e => Right_id L e))).
issig (BuildIsId L e) (@id_is_left G L e) (@id_is_right G L e).
apply @trunc_sigma.
apply trunc_forall.
intros;apply trunc_forall.
Defined.
Lemma left_id_id : forall {G} {L : Gop G} {Hg : Commutative L} (e : G),
Left_id L e -> IsId L e.
Proof.
intros ? ? ? ? X. split;auto.
intro.
eapply concat. apply commutative. apply _. apply X.
Defined.
Definition identity_sig : forall {G} (L : Gop G), sigT (IsId L) <~> Identity L.
Proof.
intros; issig (BuildIdentity L) (@g_id G L) (@g_idP G L).
Defined.
Instance gidPop : forall {G} {L : Gop G} {Hset : IsHSet G},
IsHProp (Identity L).
Proof.
intros G L Hset. eapply trunc_equiv'. apply identity_sig.
apply hprop_allpath. intros [x Hx] [y Hy].
apply path_sigma with (id_unique _ _ _ _ _).
apply id_prop.
Defined.
Definition ismonoid_sig : forall {G} (L : Gop G), sigT (fun _ : IsSemigroup L =>
Identity L) <~> IsMonoid L.
Proof.
intros;issig (BuildIsMonoid L) (@monoid_is_sg G L) (@monoid_id G L).
Defined.
Instance ismonoid_prop : forall {G} {L : Gop G} {Hset : IsHSet G},
IsHProp (IsMonoid L).
Proof.
intros.
eapply trunc_equiv'. apply ismonoid_sig.
apply @trunc_sigma. apply _.
intro. apply _.
Defined.
Lemma left_cancel_cancel : forall {G} {L : Gop G} {Hcomm : Commutative L},
forall a : G, Lcancel L a -> Cancel L a.
Proof.
intros ? ? ? ? X. split;auto.
red;intros ? ? X0.
red in X. apply X.
eapply concat. apply Hcomm. eapply concat. apply X0. apply Hcomm.
Defined.
Instance cancelling_prop : forall {G} {L : Gop G} {Hset : IsHSet G} a,
IsHProp (Cancel L a).
Proof.
intros.
apply (@trunc_equiv' (sigT (fun _ : Lcancel L a => Rcancel L a))).
issig (BuildCancel _ a) (@Left_cancel _ _ a) (@Right_cancel _ _ a).
apply @trunc_sigma;[|intro C;clear C];apply trunc_forall.
Defined.
Instance iscmono_prop : forall {G} {L : Gop G} {Hset : IsHSet G},
IsHProp (IsCMonoid L).
Proof.
intros.
apply (@trunc_equiv' (sigT (fun _ : IsMonoid L => forall a : G, Cancel L a))).
issig (BuildIsCMonoid L) (@cmonoid_is_monoid G L) (@cmonoid_cancel G L).
apply trunc_sigma.
Defined.
Lemma linverse_inverse : forall {G} {L : Gop G} {Hg : Commutative L} (x y : G),
Linverse L x y -> IsInverse L x y.
Proof.
intros;split;auto.
red. apply transport with (gop x y). apply Hg.
apply _.
Defined.
Lemma linverse_eq_gid : forall {G} {L : Gop G} {Hg : IsMonoid L}
(x y : G) {Hinv : Linverse L x y},
gop x y = gidV.
Proof.
intros. eapply id_unique;apply _.
Defined.
Lemma rinverse_eq_gid : forall {G} {L : Gop G} {Hg : IsMonoid L}
(x y : G) {Hinv : Rinverse L x y},
gop y x = gidV.
Proof.
intros;eapply id_unique;apply _.
Defined.
Lemma isinverse_sig : forall {G} {L : Gop G} x y,
sigT (fun _ : Linverse L x y => Rinverse L x y) <~> IsInverse L x y.
Proof.
intros.
issig (BuildIsInverse L x y) (@inverse_left G L x y) (@inverse_right G L x y).
Defined.
Instance isinverse_prop : forall {G} {L : Gop G} {Hset : IsHSet G} x y,
IsHProp (IsInverse L x y).
Proof.
intros.
eapply trunc_equiv'. apply isinverse_sig.
apply @trunc_sigma;intros;apply id_prop.
Defined.
Instance inverse_symm : forall {G} (L : Gop G), Symmetric (IsInverse L).
Proof.
red. intros ? ? ? ? [? ?]. split;red;apply _.
Defined.
Lemma inverse_symm_rw G L x y : @IsInverse G L x y = IsInverse L y x.
Proof.
apply isequiv_equiv_path.
eapply transitive_equiv;[|apply isinverse_sig].
apply symmetric_equiv.
issig (fun H H' => BuildIsInverse L x y H' H)
(@inverse_right G L x y) (@inverse_left G L x y).
Defined.
Instance rinverse_lcancel : forall {G} {L : Gop G} {Hassoc : Associative L},
forall a b : G, Rinverse L a b -> Lcancel L a.
Proof.
intros ? ? ? ? ? H.
red. intros x y ?.
path_via (gop (gop b a) x). apply inverse;apply H.
path_via (gop b (gop a x)).
path_via (gop b (gop a y)). apply ap;assumption.
path_via (gop (gop b a) y).
apply H.
Defined.
Instance linverse_rcancel : forall {G} {L : Gop G} {Hassoc : Associative L},
forall a b : G, Linverse L a b -> Rcancel L a.
Proof.
intros ? ? ? ? ? H.
red. intros x y ?.
path_via (gop x (gop a b)). apply inverse;apply H.
path_via (gop (gop x a) b).
path_via (gop (gop y a) b). apply (ap (fun g => gop g b) X).
path_via (gop y (gop a b)).
apply H.
Defined.
Instance isinverse_cancel : forall {G} {L : Gop G} {Hassoc : Associative L},
forall a b : G, IsInverse L a b -> Cancel L a.
Proof.
intros;split;apply _.
Defined.
Instance inverse_cancel : forall {G} {L : Gop G} {Hassoc : Associative L}
(a : G), Inverse L a -> Cancel L a.
Proof.
intros ? ? ? ? [b H]. apply isinverse_cancel with b.
assumption.
Defined.
Definition easyIsGroup {G} (L : Gop G) {Hg : IsMonoid L}
(gopp : forall x : G, Inverse L x) :=
BuildIsGroup L (BuildIsCMonoid L Hg
(fun a => inverse_cancel a (gopp a))) gopp.
Lemma inverse_unicity : forall {G} {L : Gop G} {Hassoc : Associative L},
forall x : G, atmost1P (IsInverse L x).
Proof.
intros ? ? ? ? y y' X X0. apply inverse_cancel with x. exists y;exact X.
eapply id_unique;apply _.
Defined.
Definition inverse_sig : forall {G} {L : Gop G} (x : G),
sigT (IsInverse L x) <~> Inverse L x.
Proof.
intros.
issig (BuildInverse L x) (@inverse_val G L x) (@inverse_pr G L x).
Defined.
Instance inverse_prop : forall {G} {L : Gop G} {Hset : IsHSet G}
{Ha : Associative L} (x : G), IsHProp (Inverse L x).
Proof.
intros.
eapply trunc_equiv'. apply inverse_sig.
apply hprop_allpath. intros [y H] [y' Hy'].
apply path_sigma with (inverse_unicity _ _ _ _ _). apply isinverse_prop.
Defined.
Lemma group_inverse_gopp : forall {G} {L : Gop G} {Hg : IsGroup L},
forall x y : G, IsInverse L x y -> (goppV x) = y.
Proof.
intros.
apply (@inverse_unicity G L _) with x.
apply gopp.
assumption.
Defined.
Lemma gopp_gopp : forall {G} {L : Gop G} {Hg : IsGroup L} (x : G),
(goppV (goppV x)) = x.
Proof.
intros. apply group_inverse_gopp.
apply inverse_symm. apply gopp.
Defined.
Instance gid_isinverse : forall {G} {L : Gop G} {Hg : IsMonoid L},
IsInverse L gidV gidV.
Proof.
intros.
split;red;(apply transport with gidV;[apply symmetry;apply gid|]);try apply _.
Defined.
Lemma gopp_gid : forall {G} {L : Gop G} {Hg : IsGroup L},
(goppV gidV) = gidV.
Proof.
intros. apply group_inverse_gopp. apply _.
Defined.
Lemma linverse_gop : forall {G} {L : Gop G} {Hassoc : Associative L},
forall x x' : G, Linverse L x x' ->
forall y y' : G, Linverse L y y' ->
Linverse L (gop x y) (gop y' x').
Proof.
intros ? ? ? ? ? Hx ? ? Hy.
red. split;red;intro z.
- path_via (gop x (gop (gop y y') (gop x' z))).
eapply concat. symmetry;apply Hassoc.
eapply concat. symmetry;apply Hassoc. apply ap.
eapply concat;[|apply Hassoc]. apply ap. symmetry;apply Hassoc.
path_via (gop x (gop x' z)). apply ap. apply Hy.
path_via (gop (gop x x') z). apply Hx.
- path_via (gop (gop z x) (gop (gop y y') x')).
eapply concat;[|apply Hassoc]. apply ap.
eapply concat;[symmetry;apply Hassoc|]. apply ap. apply Hassoc.
path_via (gop (gop z x) x'). apply ap. apply Hy.
path_via (gop z (gop x x')). apply Hx.
Defined.
Lemma rinverse_gop : forall {G} {L : Gop G} {Hassoc : Associative L},
forall x x' : G, Rinverse L x x' ->
forall y y' : G, Rinverse L y y' ->
Rinverse L (gop x y) (gop y' x').
Proof.
intros ? ? ?.
change (forall x x' : G, Linverse L x' x -> forall y y' : G, Linverse L y' y ->
Linverse L (gop y' x') (gop x y)).
intros;apply linverse_gop;assumption.
Defined.
Lemma isinverse_gop : forall {G} {L : Gop G} {Hassoc : Associative L},
forall x x' : G, IsInverse L x x' ->
forall y y' : G, IsInverse L y y' ->
IsInverse L (gop x y) (gop y' x').
Proof.
intros;split;[apply linverse_gop|apply rinverse_gop];apply _.
Defined.
Lemma gopp_gop : forall {G} {L : Gop G} {Hg : IsGroup L}, forall x y : G,
(goppV (gop x y)) = (gop (goppV y) (goppV x)).
Proof.
intros. apply group_inverse_gopp.
apply isinverse_gop;apply _.
Defined.
Lemma gopp_cop_comm : forall {G} {L : Gop G} {Hg : IsGroup L}, forall x y : G,
(goppV (gop x y)) = (gop (goppV x) (goppV y)).
Proof.
intros;path_via (gop (goppV y) (goppV x)).
apply gopp_gop. apply commutative. apply _.
Defined.
Instance gopp_prop : forall {G} {L : Gop G} {Hg : Associative L}
{Hset : IsHSet G}, IsHProp (forall x : G, Inverse L x).
Proof.
intros.
apply trunc_forall.
Defined.
Definition isgroup_sig : forall {G} {L : Gop G},
sigT (fun Hmon : IsCMonoid L => forall x : G, Inverse L x) <~> IsGroup L.
Proof.
intros.
issig (BuildIsGroup L) (@group_is_cmonoid G L) (@gopp G L).
Defined.
Instance isgroup_prop : forall {G} {L : Gop G} {Hset : IsHSet G},
IsHProp (IsGroup L).
Proof.
intros;eapply trunc_equiv'. apply isgroup_sig.
apply @trunc_sigma. apply _.
intro. apply _.
Defined.
End Magma_pr.
Module Relation_pr.
Export Relation.
Import HoTT.Basics hit.Truncations.
Generalizable Variable T.
Instance trans_prop : forall {T} {r : Rel T}
{Hprop : RelationProp r}, IsHProp (Transitive r).
Proof.
intros.
apply trunc_forall.
Defined.
Instance refl_prop : forall {T} {r : Rel T}
{Hprop : RelationProp r}, IsHProp (Reflexive r).
Proof.
intros.
apply trunc_forall.
Defined.
Instance symm_prop : forall {T} {r : Rel T}
{Hprop : RelationProp r}, IsHProp (Symmetric r).
Proof.
intros.
apply trunc_forall.
Defined.
Definition IsEquivalence_sig : forall {T} {r : Rel T},
sigT (fun _ : Reflexive r => sigT (fun _ : Symmetric r => Transitive r))
<~> Equivalence r.
Proof.
intros ? r. issig (BuildEquivalence _ r) (@equivalence_refl _ r)
(@equivalence_symm _ r) (@equivalence_trans _ r).
Defined.
Instance isequivalence_prop : forall {T} {r : Rel T}
{Hprop : RelationProp r}, IsHProp (Equivalence r).
Proof.
intros.
eapply trunc_equiv'.
apply IsEquivalence_sig.
apply @trunc_sigma. apply _.
intro. apply @trunc_sigma. apply _.
intro;apply _.
Defined.
Lemma irrefl_neq : forall {T} {r : Rel T} {Hr : Irreflexive r},
forall x y : T, rrel x y -> neq x y.
Proof.
intros ? ? ? ? ? ?. intros H. destruct H. apply Hr with x. assumption.
Defined.
Instance antisymm_prop : forall {T} {r : Leq T} {Hset : IsHSet T},
IsHProp (Antisymmetric r).
Proof.
intros.
apply trunc_forall.
Defined.
Instance irrefl_prop : forall {T} {r : Rel T}, IsHProp (Irreflexive r).
Proof.
intros.
apply trunc_forall.
Defined.
Definition poset_sig : forall {T} {r : Leq T}, sigT (fun _ : Transitive r =>
sigT (fun _ : Reflexive r => Antisymmetric r)) <~> Poset r.
Proof.
intros;
issig (BuildPoset _ r) (@order_trans _ r) (@order_refl _ r)
(@order_antisymm _ r).
Defined.
Instance poset_prop : forall {T} {r : Leq T} {Hset : IsHSet T}
{Hprop : RelationProp r}, IsHProp (Poset r).
Proof.
intros.
eapply trunc_equiv'.
apply poset_sig.
apply @trunc_sigma. apply trans_prop.
intro. apply @trunc_sigma. apply refl_prop.
intro;apply _.
Defined.
Instance cotrans_prop : forall {T} {r : Rel T}, IsHProp (Cotransitive r).
Proof.
intros.
apply trunc_forall.
Defined.
Definition apart_sig : forall {T} {r : Apart T}, sigT (fun _ : Irreflexive r =>
sigT (fun _ : Symmetric r => sigT (fun _ : Cotransitive r =>
forall x y : T, ~ x # y -> x=y))) <~> Apartness r.
Proof.
intros.
issig (BuildApartness _ r) (@apart_irrefl _ r) (@apart_symm _ r)
(@apart_cotrans _ r) (@apart_tight _ r).
Defined.
Instance apart_prop : forall {T} {r : Apart T} {Hset : IsHSet T}
{Hpr : RelationProp r}, IsHProp (Apartness r).
Proof.
intros;eapply trunc_equiv'.
apply apart_sig.
apply @trunc_sigma. apply irrefl_prop.
intro. apply @trunc_sigma. apply symm_prop.
intro. apply @trunc_sigma. apply cotrans_prop.
intro. apply trunc_forall.
Defined.
Lemma neq_not_not_apart : forall {T} {r : Apart T} {Ha : Apartness r},
forall x y : T, ~x=y -> ~ ~ x # y.
Proof.
intros ? ? ? ? ? H ?.
apply H;apply apart_tight. assumption.
Defined.
Lemma apart_prove_eq : forall `{r : Apart T} {Ha : Apartness r},
forall a b x y : T, (x # y -> a=b) -> (~x=y -> a=b).
Proof.
intros ? ? ? ? ? ? ? H H'.
apply apart_tight. intro H0.
assert (Hx : ~ ~ x # y).
apply neq_not_not_apart. assumption.
apply Hx;clear Hx;intro Hx.
apply irrefl_neq in H0. apply H0. auto.
Defined.
Instance islinear_prop : forall `{r : Leq T}, IsHProp (Linear r).
Proof.
intros.
apply trunc_forall.
Defined.
Instance constrlinear_linear : forall `{r : Leq T},
ConstrLinear r -> Linear r.
Proof.
red;intros.
apply tr;auto.
Defined.
Lemma dec_linear_constrlinear : forall `(r : Leq T) {Hdec : Decidable r}
{Hlin : Linear r}, ConstrLinear r.
Proof.
red;intros.
destruct (Hdec x y). left;assumption.
destruct (Hdec y x). right;assumption.
apply Empty_rect. pose (H := Hlin x y). clearbody H;revert H.
apply Trunc_rec.
intros [H|H];auto.
Defined.
Instance constrtotal_total : forall `{r : Leq T},
ConstrTotalOrder r -> TotalOrder r.
Proof.
intros. constructor; apply _.
Defined.
Definition strictorder_sig : forall `{r : Lt T},
sigT (fun _ : Irreflexive r => Transitive r) <~> StrictOrder r.
Proof.
intros. issig (BuildStrictOrder T r) (@strictorder_irrefl T r)
(@strictorder_trans T r).
Defined.
Instance strictorder_prop : forall `{r : Lt T}
{Hp : RelationProp r}, IsHProp (StrictOrder r).
Proof.
intros. eapply trunc_equiv'.
apply strictorder_sig.
apply @trunc_sigma. unfold Irreflexive. apply _. intros;apply trans_prop.
Defined.
Lemma strict_poset_antisymm : forall `(r : Lt T)
{Hstrict : StrictOrder r}, forall x y : T, x < y /\ y < x -> Empty.
Proof.
intros ? ? ? ? ? [? ?]. destruct Hstrict as [Hirr Htrans].
apply Hirr with x;eapply Htrans;eauto.
Defined.
Instance strict_poset_disjunct_prop : forall `(r : Lt T)
{Hp : RelationProp r} {Hstrict : StrictOrder r},
forall x y : T, IsHProp (x < y \/ y < x).
Proof.
intros. apply hprop_allpath.
intros [Hx|Hx] [Hy|Hy];try (apply ap;apply Hp).
destruct (strict_poset_antisymm _ _ _ (Hx, Hy)).
destruct (strict_poset_antisymm _ _ _ (Hx, Hy)).
Defined.
Lemma neq_irrefl : forall `(r : Apart T), (forall x y : T, apart x y -> ~ x=y) ->
Irreflexive r.
Proof.
intros ? ? H0;intros x H.
apply H0 in H. apply H;reflexivity.
Defined.
Lemma neq_symm : forall `(r : Apart T), (forall x y : T, x # y <-> ~x=y) ->
Symmetric r.
Proof.
intros ? ? H ? ? H'.
apply H. apply H in H'. intro H0;apply H';symmetry;assumption.
Defined.
Lemma neq_cotrans : forall `(r : Apart T), DecidablePaths T ->
(forall x y : T, x # y <-> ~x=y) -> Cotransitive r.
Proof.
intros ? ? Hdec Hneq x y H z.
apply tr.
destruct (Hdec x z) as [[]|H']. right. assumption.
left;auto;apply Hneq;auto.
Defined.
Lemma neq_apart : forall `(r : Apart T), DecidablePaths T ->
(forall x y : T, x # y <-> ~x=y) -> Apartness r.
Proof.
intros ? ? Hdec Hneq;split.
apply neq_irrefl;intros;apply Hneq;assumption.
apply neq_symm;assumption.
apply neq_cotrans;assumption.
intros ? ? H.
destruct (Hdec x y) as [?|n]. assumption.
apply Hneq in n. destruct H;assumption.
Defined.
Section PseudoOrder.
Context T {r : ApartLt T}
{Hpr : RelationProp (<)} {Hpo : PseudoOrder r}.
Lemma apart_total_lt (x y : T) : x # y -> x < y \/ y < x.
Proof.
apply apart_iff_total_lt.
Defined.
Lemma pseudo_order_lt_apart (x y : T) : x < y -> x # y.
Proof.
intros. apply apart_iff_total_lt. left;assumption.
Defined.
Lemma pseudo_order_lt_apart_flip (x y : T) : x < y -> y # x.
Proof.
intros. apply apart_iff_total_lt. right;assumption.
Defined.
Lemma not_lt_apart_lt_flip (x y : T) : ~x < y -> x # y -> y < x.
Proof.
intros H H'. apply apart_iff_total_lt in H'. destruct H';auto.
destruct H;assumption.
Defined.
Lemma pseudo_order_cotrans_twice (x1 y1 x2 y2 : T)
: x1 < y1 -> merely (x2 < y2 \/ x1 < x2 \/ y2 < y1).
Proof.
intros E1.
pose (Htmp := iscotransitive _ _ E1 x2).
clearbody Htmp. change ((merely ((x1 < x2) \/ (x2 < y1))): Type) in Htmp.
revert Htmp;apply Trunc_rec.
intros [E2|E2].
apply tr;auto.
pose (Htmp := iscotransitive _ _ E2 y2). clearbody Htmp.
revert Htmp;apply Trunc_rec.
intros [E3|E3];apply tr; auto.
Defined.
Lemma pseudo_order_lt_ext (x1 y1 x2 y2 : T)
: x1 < y1 -> merely (x2 < y2 \/ x1 # x2 \/ y2 # y1).
Proof.
intros E.
pose (H := pseudo_order_cotrans_twice x1 y1 x2 y2 E). clearbody H;revert H.
apply Trunc_rec.
intros [?|[?|?]];apply tr; auto using pseudo_order_lt_apart.
Defined.
Global Instance pseudo_is_strict : StrictOrder (<).
Proof.
split.
- intros x E. destruct pseudoorder_antisym with x x;assumption.
- intros x y z E1 E2.
pose (Htmp := iscotransitive _ _ E1 z);clearbody Htmp;revert Htmp.
pose (Hpr' := Hpr x z);clearbody Hpr';clear Hpr.
apply Trunc_rec. intros [?|?].
assumption.
destruct pseudoorder_antisym with y z;auto.
Defined.
Global Instance pseudo_complement_trans : Transitive (fun x y : T => ~ x<y).
Proof.
intros x y z.
intros E1 E2 E3.
pose (Htmp := iscotransitive _ _ E3 y);clearbody Htmp;revert Htmp.
apply Trunc_rec.
intros [?|?] ; contradiction.
Defined.
Global Instance pseudo_complement_antisym
: Antisymmetric (fun x y => ~ x < y).
Proof.
red;simpl;unfold leq.
intros x y H H0.
apply (@apart_tight T (#) _).
intro H';apply apart_iff_total_lt in H'. destruct H' as [H'|H'];auto.
Defined.
End PseudoOrder.
Section FullPoset.
Context T {r : FullRelation T}
{Hpr : RelationProp (#)} {Hpo : FullPoset r}.
Lemma lt_le : forall x y : T, x < y -> x <= y.
Proof.
intros ? ? H.
apply lt_iff_le_apart in H. apply H.
Defined.
Lemma lt_apart : forall (x y : T), x < y -> x # y.
Proof.
intros ? ? H.
apply lt_iff_le_apart in H. apply H.
Defined.
Lemma le_not_lt_flip : forall (x y : T), x <= y -> ~ y < x.
Proof.
intros ? ? H H'.
apply lt_iff_le_apart in H'.
destruct H' as [H0 H1].
apply @irrefl_neq in H1. apply H1.
apply (@antisymmetric _ (<=));try assumption. apply Hpo.
apply Hpo.
Defined.
Lemma lt_not_le_flip : forall (x y : T), x < y -> ~ y <= x.
Proof.
intros ? ? H H'.
apply lt_iff_le_apart in H.
destruct H as [H0 H1].
apply (@irrefl_neq _ _ Hpo) in H1. apply H1.
apply (@antisymmetric _ (<=));try assumption. apply Hpo.
Defined.
Lemma lt_le_trans : forall x y : T, x < y ->
forall z, y <= z -> x < z.
Proof.
intros ? ? H ? H'.
apply lt_iff_le_apart in H. destruct H as [H0 H1].
apply lt_iff_le_apart. split.
- eapply (@transitivity T (<=)). apply Hpo. apply H0. apply H'.
- pose (H := apart_cotrans _ _ H1 z). clearbody H.
change (merely ((x#z) + (z#y))%type : Type) in H.
revert H. red in Hpr; apply Trunc_rec.
intros [H|H]. assumption.
apply lt_apart. apply (@symmetry T (#)) in H.
apply (@transitivity _ (<) _ _ y);apply lt_iff_le_apart;split;auto.
apply Hpo.
Defined.
Lemma le_lt_trans : forall x y : T, x <= y -> forall z, y < z -> x < z.
Proof.
intros ? ? H ? H'.
apply lt_iff_le_apart in H';apply lt_iff_le_apart.
destruct H' as [H0 H1];split.
- eapply @transitivity;[apply Hpo|apply H|apply H0].
- pose (H' := apart_cotrans _ _ H1 x).
clearbody H'. change (merely ((y#x) + (x#z))%type : Type) in H'. revert H'.
red in Hpr;apply Trunc_rec.
intros [H' | H'];trivial.
apply lt_apart.
apply (@transitivity _ (<) _ _ y);apply lt_iff_le_apart;split;auto.
apply @symmetry. apply Hpo. assumption.
Defined.
End FullPoset.
Section FullPseudoOrder.
Context T {r : FullRelation T}
{Hpr : RelationProp (<)} {Hpo : FullPseudoOrder r}.
Lemma not_lt_le_flip : forall x y : T, ~y<x -> x<=y.
Proof.
intros. apply le_iff_not_lt_flip. assumption.
Defined.
Instance: Poset (<=).
Proof.
split.
- intros x y z H H'.
apply le_iff_not_lt_flip.
apply le_iff_not_lt_flip in H.
apply le_iff_not_lt_flip in H'.
revert H;revert H'.
apply pseudo_complement_trans; apply _.
- intros x. apply le_iff_not_lt_flip.
revert x. change (Irreflexive (<)). apply @strictorder_irrefl.
apply (@pseudo_is_strict _ _ _ Hpo).
- intros x y H H'.
apply le_iff_not_lt_flip in H;apply le_iff_not_lt_flip in H'.
apply (@pseudo_complement_antisym _ (# <) _);assumption.
Defined.
Global Instance fullpseudo_is_fullposet : FullPoset r.
Proof.
split;try apply _.
- apply (@pseudo_is_strict _ _ _ Hpo).
- intros x y. split.
intro H. split. apply le_iff_not_lt_flip.
red. apply pseudoorder_antisym;assumption.
apply apart_iff_total_lt. auto.
intros [H0 H1].
apply apart_iff_total_lt in H1. destruct H1.
assumption.
apply le_iff_not_lt_flip in H0. destruct H0;assumption.
Defined.
(* since x<=y is ~y<x, the double negation is equivalent *)
Lemma le_stable : forall x y : T, ~ ~ x <= y -> x<=y.
Proof.
intros ? ? H. apply le_iff_not_lt_flip. intro.
apply H. intro H'. apply le_iff_not_lt_flip in H'. auto.
Defined.
End FullPseudoOrder.
Import OrderedRing.
Global Instance fullpseudosemi_fullpseudo T {L : PreringFull T} {Hpo : FullPseudoSemiringOrder L}
: FullPseudoOrder L.
Proof.
split;apply Hpo.
Defined.
(**TODO**)
(*
Instance maximum_is_supremum : forall r P m,
IsMaximum r P m -> IsSupremum r P m.
Proof.
intros ? ? ? [Hup Hp].
split.
intros x Hx.
apply Hx. assumption.
assumption.
Defined.
Instance minimum_is_infimum : forall r P m,
IsMinimum r P m -> IsInfimum r P m.
Proof.
intros ? ? ? [Hlow Hp].
split.
intros x Hx.
apply Hx. assumption.
assumption.
Defined.
Lemma supremum_unicity : forall r {Ho : IsPoset r},
forall P, atmost1P (IsSupremum r P).
Proof.
intros. intros m m' Hm Hm'.
apply Ho;[apply Hm;apply Hm' | apply Hm';apply Hm].
Defined.
Lemma infimum_unicity : forall r {Ho : IsPoset r},
forall P, atmost1P (IsInfimum r P).
Proof.
intros. intros m m' Hm Hm'.
apply Ho;[apply Hm';apply Hm | apply Hm;apply Hm'];apply _.
Defined.
Lemma maximum_unicity : forall r {Ho : IsPoset r},
forall P, atmost1P (IsMaximum r P).
Proof.
intros;intros m m' Hm Hm';eapply supremum_unicity;apply _.
Defined.
Lemma minimum_unicity : forall r {Ho : IsPoset r},
forall P, atmost1P (IsMinimum r P).
Proof.
intros;intros m m' Hm Hm';eapply infimum_unicity;apply _.
Defined.
Definition maximum_supremum : forall {r P}, Maximum r P -> Supremum r P.
Proof.
intros ? ? m. exists (maximum_val m). apply maximum_is_supremum;apply _.
Defined.
Definition minimum_infimum : forall {r P}, Minimum r P -> Infimum r P.
Proof.
intros ? ? m. exists (minimum_val m). apply minimum_is_infimum;apply _.
Defined.
Instance upper_prop : forall (r : LeqRelation)
{Hp : RelationProp r},
forall P m, IsHProp (IsUpper r P m).
Proof.
intros. red in Hp.
apply trunc_forall.
Defined.
Instance lower_prop : forall (r : LeqRelation)
{Hp : RelationProp r},
forall P m, IsHProp (IsLower r P m).
Proof.
intros. red in Hp.
apply trunc_forall.
Defined.
Definition ismax_sig : forall r P m, sigT (fun _ : IsUpper r P m =>
P m) <~> IsMaximum r P m.
Proof.
intros. issig (BuildIsMaximum r P m) (@maximum_upper r P m)
(@maximum_verifies r P m).
Defined.
Definition ismin_sig : forall r P m, sigT (fun _ : IsLower r P m =>
P m) <~> IsMinimum r P m.
Proof.
intros. issig (BuildIsMinimum r P m) (@minimum_lower r P m)
(@minimum_verifies r P m).
Defined.
Instance ismaximum_prop : forall (r : LeqRelation)
{Hp : RelationProp r} P {Hp' : forall x, IsHProp (P x)} m,
IsHProp (IsMaximum r P m).
Proof.
intros;eapply trunc_equiv'. apply ismax_sig.
apply trunc_sigma.
Defined.
Instance isminimum_prop : forall (r : LeqRelation)
{Hp : RelationProp r} P {Hp' : forall x, IsHProp (P x)} m,
IsHProp (IsMinimum r P m).
Proof.
intros;eapply trunc_equiv'. apply ismin_sig.
apply trunc_sigma.
Defined.
Instance issupremum_prop : forall (r : LeqRelation)
{Hp : RelationProp r},
forall P m, IsHProp (IsSupremum r P m).
Proof.
intros. apply isminimum_prop;apply _.
Defined.
Instance isinfimum_prop : forall (r : LeqRelation)
{Hp : RelationProp r},
forall P m, IsHProp (IsInfimum r P m).
Proof.
intros;apply ismaximum_prop;apply _.
Defined.
Definition maximum_sig : forall r P, sigT (IsMaximum r P) <~> Maximum r P.
Proof.
intros;issig (BuildMaximum r P) (@maximum_val r P) (@maximum_is_maximum r P).
Defined.
Definition minimum_sig : forall r P, sigT (IsMinimum r P) <~> Minimum r P.
Proof.
intros;issig (BuildMinimum r P) (@minimum_val r P) (@minimum_is_minimum r P).
Defined.
Instance maximum_prop : forall {r} {Ho : IsPoset r}
{Hr : RelationProp r} P {Hp : forall x, IsHProp (P x)},
IsHProp (Maximum r P).
Proof.
intros;eapply trunc_equiv'. apply maximum_sig.
apply hprop_allpath. intros [x Hx] [y Hy].
apply path_sigma with (maximum_unicity _ _ _ _ _ _).
apply ismaximum_prop;apply _.
Defined.
Instance minimum_prop : forall {r} {Ho : IsPoset r}
{Hr : RelationProp r} P {Hp : forall x, IsHProp (P x)},
IsHProp (Minimum r P).
Proof.
intros;eapply trunc_equiv'. apply minimum_sig.
apply hprop_allpath. intros [x Hx] [y Hy].
apply path_sigma with (minimum_unicity _ _ _ _ _ _).
apply isminimum_prop;apply _.
Defined.
Instance supremum_prop : forall {r} {Ho : IsPoset r}
{Hp : RelationProp r},
forall P, IsHProp (Supremum r P).
Proof.
intros.
apply minimum_prop;apply _.
Defined.
Instance infimum_prop : forall {r} {Ho : IsPoset r}
{Hp : RelationProp r},
forall P, IsHProp (Infimum r P).
Proof.
intros;apply maximum_prop;apply _.
Defined.
Definition islattice_sig : forall r,
sigT (fun _ : IsPoset r => sigT (fun _ :
(forall a b, Supremum r (doubleton a b)) =>
forall a b, Infimum r (doubleton a b))) <~> IsLattice r.
Proof.
intros. issig (BuildIsLattice r) (@lattice_is_poset r) (@lattice_has_sup2 r)
(@lattice_has_inf2 r).
Defined.
Instance islattice_prop : forall (r : Relation)
{Hset : IsHSet r} {Hp : RelationProp r},
IsHProp (IsLattice r).
Proof.
intros. eapply trunc_equiv'. apply islattice_sig.
repeat (apply @trunc_sigma;[try apply _ | intro]).
pose (@supremum_prop r _).
apply trunc_forall.
pose (@infimum_prop r _).
apply trunc_forall.
Defined.
Instance doubleton_is_prop : forall {T} {a b x : T}, IsHProp (doubleton a b x).
Proof.
intros;apply minus1Trunc_is_prop.
Defined.
Definition doubleton_comm_fun : forall {T} (a b x : T),
doubleton a b x -> doubleton b a x.
Proof.
intros ? ? ? ?;apply minus1Trunc_rect_nondep.
intros [H|H];apply min1;auto.
apply minus1Trunc_is_prop.
Defined.
Lemma doubleton_comm_equiv : forall {T} (a b x : T),