From: Adam Megacz Date: Tue, 5 Apr 2011 06:33:20 +0000 (+0000) Subject: PreMonoidalCategories: remove the very last [[admit]] X-Git-Url: http://git.megacz.com/?p=coq-categories.git;a=commitdiff_plain;h=5f3bdb7947de02d8d60f1af77c999a3c80f7dbba PreMonoidalCategories: remove the very last [[admit]] --- diff --git a/src/PreMonoidalCategories.v b/src/PreMonoidalCategories.v index e56c727..07513d8 100644 --- a/src/PreMonoidalCategories.v +++ b/src/PreMonoidalCategories.v @@ -156,8 +156,8 @@ Class PreMonoidalFunctor ; mf_center : forall `(f:a~>b), CentralMorphism f -> CentralMorphism (mf_F \ f) ; mf_cancell : ∀ b, #(pmon_cancell _) ~~ #mf_i ⋉ _ >>> #(mf_first b I1) >>> mf_F \ #(pmon_cancell b) ; mf_cancelr : ∀ a, #(pmon_cancelr _) ~~ _ ⋊ #mf_i >>> #(mf_second a I1) >>> mf_F \ #(pmon_cancelr a) -; mf_assoc : ∀ a b c, #(pmon_assoc _ _ _) >>> _ ⋊ #(mf_second _ _) >>> #(mf_second _ _) ~~ - #(mf_second _ _) ⋉ _ >>> #(mf_second _ _) >>> mf_F \ #(pmon_assoc a c b) +; mf_assoc : ∀ a b c, #(pmon_assoc _ _ _) >>> _ ⋊ #(mf_first _ _) >>> #(mf_second _ _) ~~ + #(mf_second _ _) ⋉ _ >>> #(mf_first _ _) >>> mf_F \ #(pmon_assoc a c b) }. Coercion mf_F : PreMonoidalFunctor >-> Functor. @@ -188,8 +188,7 @@ Section PreMonoidalFunctorsCompose. eapply ni_comp. apply (ni_associativity PMF12 PMF23 (- ⋉fobj23 (fobj12 a))). eapply ni_comp. - apply (ni_respects PMF12 PMF12 (PMF23 >>>> - ⋉fobj23 (fobj12 a)) (- ⋉fobj12 a >>>> PMF23)). - apply ni_id. + apply (ni_respects1 PMF12 (PMF23 >>>> - ⋉fobj23 (fobj12 a)) (- ⋉fobj12 a >>>> PMF23)). apply mf_first23. clear mf_first23. @@ -202,7 +201,7 @@ Section PreMonoidalFunctorsCompose. eapply ni_inv. eapply (ni_associativity _ PMF12 PMF23). - apply ni_respects; [ idtac | apply ni_id ]. + apply ni_respects2. apply ni_inv. apply mf_first12. Defined. @@ -215,8 +214,7 @@ Section PreMonoidalFunctorsCompose. eapply ni_comp. apply (ni_associativity PMF12 PMF23 (fobj23 (fobj12 a) ⋊-)). eapply ni_comp. - apply (ni_respects PMF12 PMF12 (PMF23 >>>> fobj23 (fobj12 a) ⋊-) (fobj12 a ⋊- >>>> PMF23)). - apply ni_id. + apply (ni_respects1 PMF12 (PMF23 >>>> fobj23 (fobj12 a) ⋊-) (fobj12 a ⋊- >>>> PMF23)). apply mf_second23. clear mf_second23. @@ -229,124 +227,129 @@ Section PreMonoidalFunctorsCompose. eapply ni_inv. eapply (ni_associativity (a ⋊-) PMF12 PMF23). - apply ni_respects; [ idtac | apply ni_id ]. + apply ni_respects2. apply ni_inv. apply mf_second12. Defined. - Lemma compose_assoc_coherence a b c : - (#((pmon_assoc (compose_mf a) (fobj23 (fobj12 c))) (compose_mf b)) >>> - compose_mf a ⋊ #((compose_mf_second b) c)) >>> + (* this proof is really gross; I will write a better one some other day *) + Lemma mf_associativity_comp : + ∀a b c : C1, + (#((pmon_assoc (compose_mf a) (compose_mf c)) (fobj23 (fobj12 b))) >>> + compose_mf a ⋊ #((compose_mf_first c) b)) >>> #((compose_mf_second a) (b ⊗ c)) ~~ - (#((compose_mf_second a) b) ⋉ fobj23 (fobj12 c) >>> - #((compose_mf_second (a ⊗ b)) c)) >>> compose_mf \ #((pmon_assoc a c) b). -(* - set (mf_assoc a b c) as x. - set (mf_assoc (fobj12 a) (fobj12 b) (fobj12 c)) as x'. - unfold functor_fobj in *. - simpl in *. - etransitivity. - etransitivity. - etransitivity. - Focus 3. - apply x'. - - apply iso_shift_left' in x'. - + (#((compose_mf_second a) b) ⋉ compose_mf c >>> + #((compose_mf_first c) (a ⊗ b))) >>> compose_mf \ #((pmon_assoc a c) b). + intros; intros. unfold compose_mf_second; simpl. + unfold compose_mf_first; simpl. + unfold functor_comp; simpl. + unfold ni_respects1. unfold functor_fobj; simpl. - set (mf_second (fobj12 b)) as m. - assert (mf_second (fobj12 b)=m). reflexivity. - destruct m; simpl. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite fmor_preserves_id. - setoid_rewrite fmor_preserves_id. - setoid_rewrite fmor_preserves_id. - setoid_rewrite right_identity. - setoid_rewrite left_identity. - setoid_rewrite left_identity. - setoid_rewrite left_identity. - - set (mf_second (fobj12 (a ⊗ b))) as m''. - assert (mf_second (fobj12 (a ⊗ b))=m''). reflexivity. - destruct m''; simpl. - unfold functor_fobj; simpl. - setoid_rewrite fmor_preserves_id. - setoid_rewrite fmor_preserves_id. - setoid_rewrite right_identity. - setoid_rewrite left_identity. - setoid_rewrite left_identity. - setoid_rewrite left_identity. - - set (mf_second (fobj12 a)) as m'. - assert (mf_second (fobj12 a)=m'). reflexivity. + + set (mf_first (fobj12 c)) as m'. + assert (mf_first (fobj12 c)=m'). reflexivity. destruct m'; simpl. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite left_identity. - setoid_rewrite left_identity. - setoid_rewrite left_identity. - setoid_rewrite right_identity. - assert (fobj23 (fobj12 a) ⋊ PMF23 \ id (PMF12 (b ⊗ c)) ~~ id _). - (* *) - setoid_rewrite H2. - setoid_rewrite left_identity. - assert ((id (fobj23 (fobj12 a) ⊗ fobj23 (fobj12 b)) ⋉ fobj23 (fobj12 c)) ~~ id _). - (* *) - setoid_rewrite H3. - setoid_rewrite left_identity. - assert (id (fobj23 (fobj12 a ⊗ fobj12 b)) ⋉ fobj23 (fobj12 c) ~~ id _). - (* *) - setoid_rewrite H4. - setoid_rewrite left_identity. - clear H4. - setoid_rewrite left_identity. - assert (id (fobj23 (fobj12 (a ⊗ b))) ⋉ fobj23 (fobj12 c) ~~ id _). - (* *) - setoid_rewrite H4. - setoid_rewrite right_identity. - clear H4. - assert ((fobj23 (fobj12 a) ⋊ PMF23 \ id (PMF12 b)) ⋉ fobj23 (fobj12 c) ~~ id _). - (* *) - setoid_rewrite H4. - setoid_rewrite left_identity. - clear H4. - unfold functor_comp in ni_commutes0; simpl in ni_commutes0. - unfold functor_comp in ni_commutes; simpl in ni_commutes. - unfold functor_comp in ni_commutes1; simpl in ni_commutes1. + set (mf_second (fobj12 a)) as m. + assert (mf_second (fobj12 a)=m). reflexivity. + destruct m; simpl. - unfold functor_fobj in *. - simpl in *. - setoid_rewrite x in x'. - rewrite H1. - set (ni_commutes0 (a ) - setoid_rewrite fmor_preserves_id. + Implicit Arguments id [[Ob][Hom][Category][a]]. + idtac. + + symmetry. etransitivity. - eapply comp_respects. + repeat setoid_rewrite <- fmor_preserves_comp. + repeat setoid_rewrite fmor_preserves_id. + repeat setoid_rewrite left_identity. + repeat setoid_rewrite right_identity. reflexivity. - eapply comp_respects. - eapply comp_respects. - apply - Focus 2. - eapply fmor_preserves_id. - setoid_rewrite (fmor_preserves_id PMF23). -*) - admit. - Qed. + symmetry. + etransitivity. + repeat setoid_rewrite <- fmor_preserves_comp. + repeat setoid_rewrite fmor_preserves_id. + repeat setoid_rewrite left_identity. + repeat setoid_rewrite right_identity. + reflexivity. + + assert ( (#((pmon_assoc (fobj23 (fobj12 a)) (fobj23 (fobj12 c))) + (fobj23 (fobj12 b))) >>> + fobj23 (fobj12 a) + ⋊ ( + (#(ni_iso (fobj12 b)) >>> ( (PMF23 \ #((mf_first c) b) ))))) >>> + ( + (#(ni_iso0 (fobj12 (b ⊗ c))) >>> + ((PMF23 \ #((mf_second a) (b ⊗ c)))))) ~~ + (( + (#(ni_iso0 (fobj12 b)) >>> ( (PMF23 \ #((mf_second a) b) )))) + ⋉ fobj23 (fobj12 c) >>> + ( + (#(ni_iso (fobj12 (a ⊗ b))) >>> + ( (PMF23 \ #((mf_first c) (a ⊗ b))))))) >>> + PMF23 \ (PMF12 \ #((pmon_assoc a c) b)) + ). + repeat setoid_rewrite associativity. + setoid_rewrite (fmor_preserves_comp PMF23). + unfold functor_comp in *. + unfold functor_fobj in *. + simpl in *. + rename ni_commutes into ni_commutes7. + set (mf_assoc(PreMonoidalFunctor:=PMF12)) as q. + set (ni_commutes7 _ _ (#((mf_second a) b))) as q'. + simpl in q'. + repeat setoid_rewrite associativity. + symmetry. + setoid_rewrite <- (fmor_preserves_comp (-⋉ fobj23 (fobj12 c))). + repeat setoid_rewrite <- associativity. + setoid_rewrite juggle1. + setoid_rewrite <- q'. + repeat setoid_rewrite associativity. + setoid_rewrite fmor_preserves_comp. + idtac. + unfold functor_fobj in *. + simpl in *. + repeat setoid_rewrite <- associativity. + setoid_rewrite <- q. + clear q. + repeat setoid_rewrite <- fmor_preserves_comp. + repeat setoid_rewrite <- associativity. + apply comp_respects; try reflexivity. + + set (mf_assoc(PreMonoidalFunctor:=PMF23) (fobj12 a) (fobj12 b) (fobj12 c)) as q. + unfold functor_fobj in *. + simpl in *. + + rewrite H in q. + rewrite H0 in q. + simpl in q. + repeat setoid_rewrite <- associativity. + repeat setoid_rewrite <- associativity in q. + setoid_rewrite <- q. + clear q. + unfold functor_fobj; simpl. + + repeat setoid_rewrite associativity. + apply comp_respects; try reflexivity. + apply comp_respects; try reflexivity. + auto. + + repeat setoid_rewrite associativity. + repeat setoid_rewrite associativity in H1. + repeat setoid_rewrite <- fmor_preserves_comp in H1. + repeat setoid_rewrite associativity in H1. + apply H1. + Qed. + Implicit Arguments id [[Ob][Hom][Category]]. + + (* this proof is really gross; I will write a better one some other day *) Instance PreMonoidalFunctorsCompose : PreMonoidalFunctor PM1 PM3 (fobj23 ○ fobj12) := { mf_i := compose_mf_i ; mf_F := compose_mf ; mf_first := compose_mf_first ; mf_second := compose_mf_second }. + intros; unfold compose_mf_first; unfold compose_mf_second. set (mf_first (PMF12 a)) as x in *. set (mf_second (PMF12 b)) as y in *. @@ -359,31 +362,12 @@ Section PreMonoidalFunctorsCompose. repeat setoid_rewrite right_identity. set (mf_consistent (PMF12 a) (PMF12 b)) as later. apply comp_respects; try reflexivity. - unfold functor_comp. - unfold functor_fobj; simpl. - set (ni_commutes _ _ (id (fobj12 b))) as x. - unfold functor_comp in x. - simpl in x. - unfold functor_fobj in x. - symmetry in x. - etransitivity. - apply x. - clear x. - set (ni_commutes0 _ _ (id (fobj12 a))) as x'. - unfold functor_comp in x'. - simpl in x'. - unfold functor_fobj in x'. - etransitivity; [ idtac | apply x' ]. - clear x'. - setoid_rewrite fmor_preserves_id. - setoid_rewrite fmor_preserves_id. - setoid_rewrite right_identity. rewrite <- H in later. rewrite <- H0 in later. simpl in later. apply later. apply fmor_respects. - apply (mf_consistent a b). + apply mf_consistent. intros. simpl. @@ -419,22 +403,13 @@ Section PreMonoidalFunctorsCompose. repeat setoid_rewrite associativity. apply comp_respects; try reflexivity. - set (ni_commutes _ _ (id (fobj12 I1))) as x. + set (ni_commutes _ _ #mf_i) as x. unfold functor_comp in x. unfold functor_fobj in x. simpl in x. - setoid_rewrite <- x. - clear x. - setoid_rewrite fmor_preserves_id. - setoid_rewrite fmor_preserves_id. - setoid_rewrite right_identity. - rewrite H. simpl. - clear H. - unfold functor_comp in ni_commutes. - simpl in ni_commutes. - apply ni_commutes. + apply x. intros. unfold compose_mf_second; simpl. @@ -464,24 +439,16 @@ Section PreMonoidalFunctorsCompose. repeat setoid_rewrite associativity. apply comp_respects; try reflexivity. - set (ni_commutes _ _ (id (fobj12 I1))) as x. + set (ni_commutes _ _ #mf_i) as x. unfold functor_comp in x. unfold functor_fobj in x. simpl in x. - setoid_rewrite <- x. - clear x. - setoid_rewrite fmor_preserves_id. - setoid_rewrite fmor_preserves_id. - setoid_rewrite right_identity. - rewrite H. simpl. - clear H. - unfold functor_comp in ni_commutes. - simpl in ni_commutes. - apply ni_commutes. + apply x. + + apply mf_associativity_comp. - apply compose_assoc_coherence. Defined. End PreMonoidalFunctorsCompose. @@ -806,4 +773,4 @@ Section PreMonoidalFullSubcategory. }. Defined. End PreMonoidalFullSubcategory. -*) \ No newline at end of file +*)