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Lemma τ_lmodule_laws : LModule_Mor_laws T_mon (T:=HT_mod) (T' := T_mod) (τ T). Proof. intro a. apply pathsinv0. exact (nat_trans_eq_pointwise (fbracket_τ T (Z:= p T)(identity _ )) a). Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.CategoryTheory.Monads.LModules. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.GenMendlerIteration_alt. Require Import UniMath.CategoryTheory.Limits.Initial. | SubstitutionSystems\SimplifiedHSS\ModulesFromSignatures.v | τ_lmodule_laws | 38,200 |
Definition τ_lmodule_mor : LModule_Mor _ _ _ := tpair (λ x, LModule_Mor_laws _ x) _ τ_lmodule_laws. End TauModuleMorphism. Context (IC : Initial C) (CC : Colims_of_shape nat_graph C) (HH : is_omega_cocont H). Let T := InitHSS _ CP IC CC H HH. Local Notation T_alg := (alg_from_hetsubst _ _ _ (hetsubst_from_hss _ _ _ T)). Local Notation T_mon := (Monad_from_hss _ _ _ T). Local Notation T_func := (T_mon : functor _ _). Local Notation T_hss := (T:hss _ _). Section fix_a_representation. Context (M : Monad C). Local Notation M_mod := (tautological_LModule M). Local Notation HM_mod := (lift_lmodule H _ M_mod). Context (τ_M: LModule_Mor M HM_mod M_mod). Local Definition M_alg : Alg. Proof. apply (tpair (λ x, EndC ⟦ Id_H x, x ⟧) (M:functor _ _)). apply BinCoproductArrow. - apply Monads.η. - apply τ_M. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.CategoryTheory.Monads.LModules. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.GenMendlerIteration_alt. Require Import UniMath.CategoryTheory.Limits.Initial. | SubstitutionSystems\SimplifiedHSS\ModulesFromSignatures.v | τ_lmodule_mor | 38,201 |
Lemma j_mor_rep x : τT x · j_mor x = (# H j_mor:nat_trans _ _) x · τ_M x. Proof. etrans; [ apply assoc' |]. etrans. { apply cancel_precomposition. apply (nat_trans_eq_pointwise (algebra_mor_commutes _ _ _ j) x). } etrans; [ apply assoc |]. etrans. { apply cancel_postcomposition. apply BinCoproductIn2Commutes. } etrans; [ apply assoc' |]. apply cancel_precomposition. apply BinCoproductIn2Commutes. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.CategoryTheory.Monads.LModules. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.GenMendlerIteration_alt. Require Import UniMath.CategoryTheory.Limits.Initial. | SubstitutionSystems\SimplifiedHSS\ModulesFromSignatures.v | j_mor_rep | 38,202 |
Lemma j_mon_η : ∏ a : C, (Monads.η T_mon) a · j_mor a = (Monads.η M) a. Proof. intro a. etrans; [ apply assoc' |]. etrans. { apply cancel_precomposition. apply (nat_trans_eq_pointwise (algebra_mor_commutes _ _ _ j) a). } etrans; [ apply assoc |]. etrans. { apply cancel_postcomposition. apply BinCoproductIn1Commutes. } etrans; [ apply assoc' |]. etrans. { apply cancel_precomposition. apply BinCoproductIn1Commutes. } apply id_left. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.CategoryTheory.Monads.LModules. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.GenMendlerIteration_alt. Require Import UniMath.CategoryTheory.Limits.Initial. | SubstitutionSystems\SimplifiedHSS\ModulesFromSignatures.v | j_mon_η | 38,203 |
Lemma j_mon_square_eq1 : # L (alg_map Id_H T_alg) · ((μ T_mon : EndC ⟦_, _⟧) · j_mor) = (ψ : nat_trans _ _) `T_alg ((μ T_mon : EndC ⟦_, _⟧) · j_mor). Proof. apply coprod_iter_eq; intro x. - etrans; [ apply assoc |]. etrans. { apply cancel_postcomposition. apply (Monad_law1 (T:=T_mon)). } apply id_left. - etrans; [ apply assoc |]. etrans. { apply cancel_postcomposition. apply (LModule_Mor_σ _ τT). } etrans; [ apply assoc' |]. etrans; [ apply assoc' |]. etrans; [| apply assoc ]. apply cancel_precomposition. rewrite functor_comp. etrans; [| apply assoc ]. apply cancel_precomposition. apply j_mor_rep. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.CategoryTheory.Monads.LModules. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.GenMendlerIteration_alt. Require Import UniMath.CategoryTheory.Limits.Initial. | SubstitutionSystems\SimplifiedHSS\ModulesFromSignatures.v | j_mon_square_eq1 | 38,204 |
Lemma j_mon_square_eq2 : # L (alg_map Id_H T_alg) · ((j_mor ø T_mon : EndC ⟦_∙_, _∙_⟧) · (M ∘ j_mor : EndC ⟦_∙_, _∙_⟧) · μ M) = (ψ : nat_trans _ _) `T_alg ((j_mor ø T_mon : EndC ⟦_∙_, _∙_⟧) · (M ∘ j_mor : EndC ⟦_∙_, _∙_⟧) · μ M). Proof. apply coprod_iter_eq; intro x. - etrans; [ apply assoc |]. etrans. { apply cancel_postcomposition. etrans; [ apply assoc |]. apply cancel_postcomposition. apply j_mon_η. } etrans. { apply cancel_postcomposition. eapply pathsinv0. apply (nat_trans_ax (Monads.η M )). } etrans; [| apply id_right ]. rewrite <- assoc. apply cancel_precomposition. apply Monad_law1. - etrans; [ apply assoc |]. etrans. { apply cancel_postcomposition. etrans; [ apply assoc |]. etrans. { apply cancel_postcomposition. apply j_mor_rep. } rewrite <- assoc. apply cancel_precomposition. eapply pathsinv0. apply (nat_trans_ax τ_M). } etrans. { repeat rewrite <- assoc. apply cancel_precomposition. apply cancel_precomposition. apply (LModule_Mor_σ _ τ_M ( x)). } repeat rewrite assoc. apply cancel_postcomposition. etrans. { repeat rewrite <- assoc. apply cancel_precomposition. etrans; [ apply assoc |]. apply cancel_postcomposition. apply (θ_nat_2_pw _ _ _ j_ptd). } etrans. { repeat rewrite assoc. apply cancel_postcomposition. apply cancel_postcomposition. apply (θ_nat_1_pw _ _ j_mor (ptd_from_mon T_mon)). } repeat rewrite <- assoc. apply cancel_precomposition. rewrite functor_comp. rewrite functor_comp. repeat rewrite assoc. apply cancel_postcomposition. rewrite <- functor_comp. etrans; [ apply functor_comp_pw |]. apply functor_cancel_pw. apply nat_trans_eq_alt. intro y. etrans. { apply cancel_postcomposition. etrans. { apply cancel_precomposition. apply functor_id. } apply id_right. } apply cancel_precomposition. apply id_left. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.CategoryTheory.Monads.LModules. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.GenMendlerIteration_alt. Require Import UniMath.CategoryTheory.Limits.Initial. | SubstitutionSystems\SimplifiedHSS\ModulesFromSignatures.v | j_mon_square_eq2 | 38,205 |
Lemma j_mon_laws : Monad_Mor_laws (T:=T_mon) (T':=M) (mor_from_algebra_mor _ j). Proof. split. - apply (nat_trans_eq_pointwise (a:= compose (C:=EndC) (μ T_mon) j_mor) (a':= compose(C:=EndC) (compose (C:=EndC) (a:=_∙_) (b:=_∙_) (c:=_∙_) (j_mor ø T_mon ) (M ∘ j_mor) ) (μ M))). apply (uniqueExists uniq_iter). + exact j_mon_square_eq1. + exact j_mon_square_eq2. - apply j_mon_η. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.CategoryTheory.Monads.LModules. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.GenMendlerIteration_alt. Require Import UniMath.CategoryTheory.Limits.Initial. | SubstitutionSystems\SimplifiedHSS\ModulesFromSignatures.v | j_mon_laws | 38,206 |
Definition j_mon : Monad_Mor T_mon M := _ ,, j_mon_laws. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.CategoryTheory.Monads.LModules. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.GenMendlerIteration_alt. Require Import UniMath.CategoryTheory.Limits.Initial. | SubstitutionSystems\SimplifiedHSS\ModulesFromSignatures.v | j_mon | 38,207 |
Definition TermHSS : hss_category CP (Presignature_Signature (H,,θ)) := InitHSS C CP IC CC (Presignature_Signature (H,,θ)) HH. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | TermHSS | 38,208 |
Definition TermHetSubst: heterogeneous_substitution C CP H := hetsubst_from_hss C CP (Presignature_Signature (H,,θ)) TermHSS. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | TermHetSubst | 38,209 |
Definition Terms: [C, C] := pr1 (pr1 TermHSS). | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | Terms | 38,210 |
Definition TermAlgebra: FunctorAlg Id_H:= pr1 TermHSS. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | TermAlgebra | 38,211 |
Definition isInitialTermAlgebra: isInitial (FunctorAlg Id_H) TermAlgebra. Proof. set (aux := colimAlgInitial InitialEndC (is_omega_cocont_Id_H C CP H HH) (Colims_of_shape_nat_graph_EndC _)). exact (pr2 aux). Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | isInitialTermAlgebra | 38,212 |
Definition TermMonad: Monad C := Monad_from_hss C CP (H,,θ) TermHSS. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | TermMonad | 38,213 |
Definition VarTerms: [C, C] ⟦ functor_identity C, Terms ⟧:= eta_from_alg TermAlgebra. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | VarTerms | 38,214 |
Definition ConstrTerms: [C, C] ⟦ H Terms, Terms ⟧ := tau_from_alg TermAlgebra. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | ConstrTerms | 38,215 |
Definition join: [C, C] ⟦ functor_compose Terms Terms, Terms ⟧ := prejoin_from_hetsubst TermHetSubst. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | join | 38,216 |
Definition joinLookup: ∏ c : C, pr1 VarTerms (pr1 Terms c) · pr1 join c = identity (pr1 Terms c) := @Monad_law1 C TermMonad. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | joinLookup | 38,217 |
Definition θforTerms := θ_from_hetsubst C CP H TermHetSubst Terms. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | θforTerms | 38,218 |
Definition joinHomomorphic: θforTerms · # H join · ConstrTerms = #(pre_composition_functor _ _ _ Terms) ConstrTerms · join := prejoin_from_hetsubst_τ TermHetSubst. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | joinHomomorphic | 38,219 |
Definition joinHasEtaLaw: ∏ c : C, # (pr1 Terms) (pr1 VarTerms c) · pr1 join c = identity (pr1 Terms c) := @Monad_law2 C TermMonad. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | joinHasEtaLaw | 38,220 |
Definition joinHasPermutationLaw: ∏ c : C, # (pr1 Terms) (pr1 join c) · pr1 join c = pr1 join (pr1 Terms c) · pr1 join c := @Monad_law3 C TermMonad. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadicSubstitution_alt.v | joinHasPermutationLaw | 38,221 |
Definition μ_0 : functor_identity C ⟹ functor_data_from_functor _ _ `T := η T. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | μ_0 | 38,222 |
Definition μ_0_ptd : id_Ptd C --> p T. Proof. exists μ_0. intro c. simpl. apply id_left. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | μ_0_ptd | 38,223 |
Definition μ_1 : functor_composite (U (id_Ptd C)) (`T) ⟹ functor_data_from_functor _ _ `T := ⦃μ_0⦄_{id_Ptd C}. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | μ_1 | 38,224 |
Lemma μ_1_identity : μ_1 = identity `T . Proof. apply pathsinv0. apply fbracket_unique. split. - apply nat_trans_eq_alt. intros; simpl. rewrite id_right. apply idpath. - apply nat_trans_eq_alt. intro c. simpl. rewrite id_right. assert (H':= θ_Strength1_int_implies_θ_Strength1 _ θ_strength1_int). red in H'. simpl in H'. assert (H2 := H' (`T)). assert (H3 := nat_trans_eq_pointwise H2 c). simpl in *. intermediate_path (identity _ · pr1 (τ T) c). + apply cancel_postcomposition. apply H3. + apply id_left. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | μ_1_identity | 38,225 |
Lemma μ_1_identity' : ∏ c : C, μ_1 c = identity _. Proof. intros c. assert (HA:= nat_trans_eq_pointwise μ_1_identity). apply HA. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | μ_1_identity' | 38,226 |
Lemma μ_1_identity_stronger : μ_1 = identity (U T). Proof. set (t':=nat_trans_eq_weq C C hs _ _ μ_1 (identity (U T))). apply invweq in t'. set (t'' := t' μ_1_identity'). exact t''. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | μ_1_identity_stronger | 38,227 |
Definition μ_2 : functor_composite (`T) (`T) ⟹ pr1 (`T) := prejoin_from_hetsubst T. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | μ_2 | 38,228 |
Definition disp_Monad_data_from_hss : disp_Monad_data `T := μ_2 ,, μ_0. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | disp_Monad_data_from_hss | 38,229 |
Lemma Monad_law_1_from_hss : ∏ c : C, μ_0 (pr1 (`T) c) · μ_2 c = identity ((pr1 (`T)) c). Proof. intro c. unfold μ_0. set (H' := prejoin_from_hetsubst_η T). set (H2:= nat_trans_eq_weq (homset_property C) _ _ H'). apply pathsinv0. apply H2. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | Monad_law_1_from_hss | 38,230 |
Lemma Monad_law_2_from_hss: ∏ c : C, # (pr1 (`T)) (μ_0 c)· μ_2 c = identity ((pr1 (`T)) c). Proof. intro c. intermediate_path (μ_1 c). - unfold μ_1. assert (H' := @fbracket_unique_target_pointwise _ _ _ T). assert (H1 := H' (id_Ptd C) μ_0). set (x := post_whisker μ_0 (`T) : EndC ⟦ `T • functor_identity _ , `T • `T ⟧). set (x' := x · μ_2). assert (H2 := H1 x'). apply H2; clear H2. unfold x'. clear x'. unfold x; clear x. clear H1. clear H'. clear c. split. + apply nat_trans_eq_alt. intro c. assert (H' := nat_trans_ax (η T)). simpl in H'. rewrite assoc. cbn. rewrite <- H'; clear H'. assert (H' := prejoin_from_hetsubst_η T). assert (H2 := nat_trans_eq_weq (homset_property C) _ _ H'). simpl in H2. rewrite <- assoc. rewrite <- H2. apply pathsinv0. apply id_right. + rewrite functor_comp. apply nat_trans_eq_alt. intro c. rewrite <- horcomp_id_postwhisker. do 2 rewrite assoc. simpl in *. unfold horcomp_data; simpl. intermediate_path ( # (pr1 (H ( (` T)))) (μ_0 c) · pr1 (θ ((`T) ⊗ (p T))) c · pr1 (# H μ_2) c · pr1 (τ T) c). * unfold tau_from_alg; cbn. do 2 rewrite assoc. do 3 apply cancel_postcomposition. assert (H' := θ_nat_2 _ _ _ H θ). assert (H2 := H' (`T) _ _ μ_0_ptd); clear H'. assert (H3 := nat_trans_eq_weq (homset_property C) _ _ H2 c); clear H2. simpl in H3. unfold horcomp_data in H3; simpl in H3. rewrite id_left in H3. apply (!H3). * assert (H' := prejoin_from_hetsubst_τ T). assert (H2 := nat_trans_eq_weq (homset_property C) _ _ H' c); clear H'. simpl in *. do 2 rewrite <- assoc. { intermediate_path ( # (pr1 (H (` T))) (μ_0 c) · (pr1 (τ T) (pr1 (`T) c) · pr1 μ_2 c)). - apply maponpaths. rewrite assoc. apply H2. - clear H2. do 2 rewrite assoc. apply cancel_postcomposition. etrans. { apply (nat_trans_ax (τ T) ). } apply cancel_postcomposition. apply pathsinv0. apply id_right. } - apply μ_1_identity'. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | Monad_law_2_from_hss | 38,231 |
Definition T_squared : Ptd := ptd_compose _ (p T) (p T). | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | T_squared | 38,232 |
Lemma μ_2_is_ptd_mor : ∏ c : C, (ptd_pt C T_squared) c · μ_2 c = pr1 (η T) c. Proof. intro c. unfold μ_2. unfold T_squared. unfold ptd_compose. rewrite functorial_composition_pre_post. simpl. assert (H' := Monad_law_2_from_hss c). simpl in H'. intermediate_path (pr1 (η T) c · identity _ ). - unfold eta_from_alg; simpl. repeat rewrite <- assoc. apply maponpaths. apply maponpaths. exact H'. - apply id_right. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | μ_2_is_ptd_mor | 38,233 |
Definition μ_2_ptd : T_squared --> p T. Proof. exists μ_2. red. apply μ_2_is_ptd_mor. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | μ_2_ptd | 38,234 |
Definition μ_3 : EndC ⟦U T_squared • `T, `T⟧ := ⦃μ_2⦄_{T_squared}. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | μ_3 | 38,235 |
Lemma μ_3_T_μ_2_μ_2 : μ_3 = (`T ∘ μ_2 : EndC ⟦`T • _ , `T • `T⟧ ) · μ_2. Proof. apply pathsinv0. apply (fbracket_unique(Z:=T_squared) T μ_2). split. - apply nat_trans_eq_alt. intro c. assert (H2 := nat_trans_ax (η T)); simpl in H2. rewrite assoc. simpl; rewrite <- H2 ; clear H2. intermediate_path (μ_2 c · identity _ ). + apply pathsinv0, id_right. + etrans; [| apply assoc ]. apply maponpaths. apply pathsinv0. apply Monad_law_1_from_hss. - rewrite functor_comp. assert (H1 := θ_nat_2 _ _ _ H θ (`T) _ _ μ_2_ptd). simpl in H1. repeat rewrite assoc. match goal with |[H1 : ?g = _ |- _ · _ · ?f · ?h = _ ] => intermediate_path (g · f · h) end. + do 2 apply cancel_postcomposition. apply pathsinv0. etrans; [apply H1 |]. clear H1. do 2 apply maponpaths. assert (H3 := horcomp_id_postwhisker). assert (H4 := H3 _ _ _ _ _ μ_2 (`T)); clear H3. apply H4. + clear H1. apply nat_trans_eq_alt. intro c; simpl. unfold horcomp_data; simpl. rewrite id_left. assert (H2 := prejoin_from_hetsubst_τ T). assert (H3 := nat_trans_eq_pointwise H2 c); clear H2. simpl in *. match goal with |[H3 : _ = ?f |- ?e · _ · _ · _ = _ ] => intermediate_path (e · f) end. * etrans; [ apply assoc' |]. etrans; [ apply assoc' |]. apply maponpaths. etrans; [| apply H3 ]. apply assoc. * clear H3. repeat rewrite assoc. apply cancel_postcomposition. assert (H1 := nat_trans_ax (τ T )). etrans; [ | apply H1]; clear H1. apply assoc'. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | μ_3_T_μ_2_μ_2 | 38,236 |
Lemma μ_3_μ_2_T_μ_2 : ( @compose (functor_category C C) T²∙T _ _ ((μ_2 •• `T) ) μ_2 : T•T² --> `T) = μ_3. Proof. apply (fbracket_unique(Z:=T_squared) T μ_2). split. - apply nat_trans_eq_alt; intro c. simpl. intermediate_path (identity _ · μ_2 c). + apply pathsinv0, id_left. + etrans; [ | apply assoc' ]. apply cancel_postcomposition. assert (H1 := Monad_law_1_from_hss (pr1 (`T) c)). apply (!H1). - set (B := τ T). match goal with | [|- _ · # ?H (?f · _ ) · _ = _ ] => set (F := f : T•T² --> _ ) end. assert (H3 := functor_comp H F μ_2). unfold functor_compose in H3. etrans. { apply cancel_postcomposition. apply maponpaths. apply H3. } clear H3. apply nat_trans_eq_alt. intro c. simpl. match goal with | [ |- ?a · _ · _ = _ ] => set (Ac := a) end. simpl in Ac. simpl in *. unfold functor_compose in *. assert (HX := θ_nat_1 _ _ _ H θ _ _ μ_2). assert (HX1 := HX (ptd_from_alg T)); clear HX. simpl in HX1. assert (HXX := nat_trans_eq_pointwise HX1 c); clear HX1. simpl in HXX. unfold horcomp_data in HXX. rewrite (functor_id ( H (`T))) in HXX. rewrite id_right in HXX. match goal with |[HXX : ?f · ?h = _ · _ |- _ · (_ · ?x ) · ?y = _ ] => intermediate_path (pr1 (θ ((`T) ⊗ (ptd_from_alg T))) (pr1 (pr1 (pr1 T)) c)· f · h · x · y) end. * repeat rewrite assoc. do 3 apply cancel_postcomposition. unfold Ac. clear Ac. etrans; [| apply assoc ]. etrans. 2: { apply maponpaths. apply (!HXX). } clear HXX. assert (Strength_2 : ∏ α : functor_compose (functor_composite (`T) (`T))(`T) --> functor_composite (` T) (`T), pr1 (θ (`T ⊗ T_squared)) c · pr1 (# H α) c = pr1 (θ ((`T) ⊗ (ptd_from_alg T))) ((pr1 (pr1 (pr1 T))) c)· pr1 (θ (( ((`T) • (`T) : [_, _])) ⊗ (ptd_from_alg T))) c· pr1 (# H (α : functor_compose (`T) (functor_composite (`T) (` T))--> _)) c ). { intro α; assert (HA := θ_Strength2_int_implies_θ_Strength2 _ θ_strength2_int); assert (HA' := HA (`T) (ptd_from_alg T) (ptd_from_alg T) _ α); clear HA; assert (HA2 := nat_trans_eq_pointwise HA' c ); clear HA'; simpl in HA2; apply HA2. } etrans; [ apply (Strength_2 F) |]. clear Strength_2. etrans; [ apply assoc' |]. do 2 apply maponpaths. match goal with |[ |- _ = ?pr1 (# ?G ?g) _ ] => assert (X : F = g) end. { apply nat_trans_eq; try apply homset_property. intros. unfold F. simpl. unfold horcomp_data; simpl. rewrite functor_id. apply pathsinv0, id_right. } apply (maponpaths (λ T, pr1 (# H T) c)). apply X. * clear HXX. clear Ac. clear F. clear B. assert (H4 := prejoin_from_hetsubst_τ T). assert (H5 := nat_trans_eq_pointwise H4 c); clear H4. simpl in H5. { match goal with |[ H5 : _ = ?e |- ?a · ?b · _ · _ · _ = _ ] => intermediate_path (a · b · e) end. - repeat rewrite <- assoc. do 2 apply maponpaths. repeat rewrite <- assoc in H5. apply H5. - clear H5. repeat rewrite assoc. apply cancel_postcomposition. assert (HT := prejoin_from_hetsubst_τ T). assert (H6 := nat_trans_eq_pointwise HT); clear HT. unfold coproduct_nat_trans_in2_data. unfold tau_from_alg, tau2_from_alg in H6. rewrite assoc in H6. apply H6. } Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | μ_3_μ_2_T_μ_2 | 38,237 |
Lemma third_monad_law_from_hss : (`T ∘ μ_2 : EndC ⟦ functor_composite (functor_composite `T `T) `T , `T • `T ⟧) · μ_2 = (rassociator_CAT _ _ _) · (μ_2 •• `T) · μ_2. Proof. intermediate_path μ_3; [apply pathsinv0, μ_3_T_μ_2_μ_2 | ]. apply pathsinv0. apply (fbracket_unique(Z:=T_squared) T μ_2). split. - apply nat_trans_eq_alt; intro c. simpl. rewrite assoc. intermediate_path (identity _ · μ_2 c). + apply pathsinv0, id_left. + apply cancel_postcomposition. rewrite id_left. assert (H1 := Monad_law_1_from_hss (pr1 (`T) c)). simpl in H1. apply (!H1). - do 2 rewrite functor_comp. do 4 rewrite assoc. unfold T_squared. apply nat_trans_eq_alt. intro c; simpl. assert (HTT := θ_strength2_int). assert (HX := HTT (`T) (ptd_from_alg T) (ptd_from_alg T)); clear HTT. assert (HX' := nat_trans_eq_pointwise HX c); clear HX. simpl in HX'. match goal with | [ H : _ = ?f |- _ · _ · ?g · ?h · ?i = _ ] => intermediate_path (f · g · h · i) end. + do 3 apply cancel_postcomposition. apply HX'. + clear HX'. rewrite id_left. rewrite id_right. assert (HX :=θ_nat_1 _ _ _ H θ _ _ μ_2). assert (HX1 := HX (ptd_from_alg T)); clear HX. simpl in HX1. assert (HXX := nat_trans_eq_pointwise HX1 c); clear HX1. simpl in HXX. unfold horcomp_data in HXX; simpl in HXX. match goal with | [ H : ?x = _ |- ?e · _ · _ · ?f · ?g = _ ] => intermediate_path (e · x · f · g) end. * do 2 apply cancel_postcomposition. repeat rewrite <- assoc. apply maponpaths. { match goal with | [ H : _ = ?x |- _ ] => intermediate_path x end. - clear HXX. apply maponpaths. match goal with | [ |- _ ?a ?x = _ ?b ?y ] => assert (TTT : a = b) end. { match goal with | [ |- _ ?a = _ ?b ] => assert (TTTT : a = b) end. { apply nat_trans_eq_alt. intros. simpl. unfold horcomp_data; simpl. rewrite functor_id. apply pathsinv0, id_right. } apply maponpaths. apply TTTT. } apply (nat_trans_eq_pointwise TTT). - repeat rewrite assoc. repeat rewrite assoc in HXX. apply (!HXX). } * clear HXX. assert (H4 := prejoin_from_hetsubst_τ T). assert (H5 := nat_trans_eq_pointwise H4 c); clear H4. unfold μ_2. repeat rewrite <- assoc. simpl in H5; repeat rewrite <- assoc in H5. etrans. { do 3 apply maponpaths. apply H5. } clear H5. rewrite functor_id. rewrite id_left. repeat rewrite assoc. apply cancel_postcomposition. assert (H4' := prejoin_from_hetsubst_τ T). assert (H6 := nat_trans_eq_pointwise H4' (pr1 `T c)); clear H4'. simpl in H6. unfold coproduct_nat_trans_in2_data in H6. simpl in H6. rewrite assoc in H6. apply H6. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | third_monad_law_from_hss | 38,238 |
Lemma disp_Monad_laws_from_hss : disp_Monad_laws disp_Monad_data_from_hss. Proof. split. - unfold disp_Monad_data_from_hss; simpl; split. + apply Monad_law_1_from_hss. + apply Monad_law_2_from_hss. - unfold disp_Monad_data_from_hss; simpl. intro c. intermediate_path (pr1 μ_3 c). + set (H1 := μ_3_T_μ_2_μ_2). set (H2 := nat_trans_eq_weq (homset_property C) _ _ H1). apply pathsinv0, H2. + set (H1 := μ_3_μ_2_T_μ_2). set (H2 := nat_trans_eq_weq (homset_property C) _ _ H1). apply pathsinv0, H2. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | disp_Monad_laws_from_hss | 38,239 |
Definition Monad_from_hss : Monad C := _ ,, _ ,, disp_Monad_laws_from_hss. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | Monad_from_hss | 38,240 |
Definition Monad_Mor_laws_from_hssMor (T T' : hss CP H)(β : hssMor T T') : Monad_Mor_laws (T:=Monad_from_hss T) (T':=Monad_from_hss T') (pr1 (pr1 β)). | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | Monad_Mor_laws_from_hssMor | 38,241 |
Definition Monad_Mor_from_hssMor {T T' : hss CP H}(β : hssMor T T') : Monad_Mor (Monad_from_hss T) (Monad_from_hss T') := tpair _ _ (Monad_Mor_laws_from_hssMor T T' β). | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | Monad_Mor_from_hssMor | 38,242 |
Definition hss_to_monad_functor_data : functor_data (hss_precategory CP H) (category_Monad C). Proof. exists Monad_from_hss. exact @Monad_Mor_from_hssMor. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | hss_to_monad_functor_data | 38,243 |
Lemma is_functor_hss_to_monad : is_functor hss_to_monad_functor_data. Proof. split; simpl. - intro a. apply (invmap (Monad_Mor_equiv _ _ )). apply idpath. - intros a b c f g. apply (invmap (Monad_Mor_equiv _ _ )). apply idpath. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | is_functor_hss_to_monad | 38,244 |
Definition hss_to_monad_functor : functor _ _ := tpair _ _ is_functor_hss_to_monad. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | hss_to_monad_functor | 38,245 |
Definition hssMor_Monad_Mor_eq {T T' : hss CP H} (β β' : hssMor T T') : β = β' ≃ Monad_Mor_from_hssMor β = Monad_Mor_from_hssMor β'. Proof. eapply weqcomp. - apply hssMor_eq. - apply invweq. use Monad_Mor_equiv. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | hssMor_Monad_Mor_eq | 38,246 |
Lemma faithful_hss_to_monad : faithful hss_to_monad_functor. Proof. unfold faithful. intros T T'. apply isinclbetweensets. - apply isaset_hssMor. - apply isaset_Monad_Mor. - intros β β'. apply (invmap (hssMor_Monad_Mor_eq _ _ )). Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. | SubstitutionSystems\SimplifiedHSS\MonadsFromSubstitutionSystems.v | faithful_hss_to_monad | 38,247 |
Definition SignatureInitialAlgebraSetSort (H : Signature HSET_over_sort HSET_over_sort HSET_over_sort) (Hs : is_omega_cocont H) : Initial (FunctorAlg (Id_H H)). Proof. use colimAlgInitial. - apply Initial_functor_precat, Initial_slice_precat, InitialHSET. - apply (is_omega_cocont_Id_H), Hs. - apply ColimsFunctorCategory_of_shape, slice_precat_colims_of_shape, ColimsHSET_of_shape. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction.v | SignatureInitialAlgebraSetSort | 38,248 |
Definition MultiSortedSigToHSS (sig : MultiSortedSig sort) : HSS (MultiSortedSigToSignature sort sig). Proof. apply SignatureToHSS. + apply Initial_slice_precat, InitialHSET. + apply slice_precat_colims_of_shape, ColimsHSET_of_shape. + apply is_omega_cocont_MultiSortedSigToSignature. apply slice_precat_colims_of_shape, ColimsHSET_of_shape. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction.v | MultiSortedSigToHSS | 38,249 |
Definition MultiSortedSigToHSSisInitial (sig : MultiSortedSig sort) : isInitial _ (MultiSortedSigToHSS sig). | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction.v | MultiSortedSigToHSSisInitial | 38,250 |
Definition MultiSortedSigToMonad (sig : MultiSortedSig sort) : Monad (HSET / sort). Proof. use Monad_from_hss. - apply BinCoproducts_HSET_slice. - apply (MultiSortedSigToSignature sort sig). - apply MultiSortedSigToHSS. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction.v | MultiSortedSigToMonad | 38,251 |
Definition SignatureInitialAlgebra (H : Signature sortToC sortToC sortToC) (Hs : is_omega_cocont H) : Initial (FunctorAlg (Id_H H)). Proof. use colimAlgInitial. - apply Initial_functor_precat, Initial_functor_precat, IC. - apply (is_omega_cocont_Id_H), Hs. - apply ColimsFunctorCategory_of_shape, ColimsFunctorCategory_of_shape, HC. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction_alt.v | SignatureInitialAlgebra | 38,252 |
Definition MultiSortedSigToHSS (sig : MultiSortedSig sort) : HSS (MultiSortedSigToSignature sort Hsort C TC BP BC CC sig). Proof. apply SignatureToHSS. + apply Initial_functor_precat, IC. + apply ColimsFunctorCategory_of_shape, HC. + apply is_omega_cocont_MultiSortedSigToSignature. - exact eqsetPC. - exact EsortToCC. - exact HC. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction_alt.v | MultiSortedSigToHSS | 38,253 |
Definition MultiSortedSigToHSSisInitial (sig : MultiSortedSig sort) : isInitial _ (MultiSortedSigToHSS sig). | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction_alt.v | MultiSortedSigToHSSisInitial | 38,254 |
Definition MultiSortedSigToMonad (sig : MultiSortedSig sort) : Monad sortToC. Proof. use Monad_from_hss. - apply BCsortToC. - apply (MultiSortedSigToSignature sort Hsort C TC BP BC CC sig). - apply MultiSortedSigToHSS. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction_alt.v | MultiSortedSigToMonad | 38,255 |
Definition projSortToSet : sort → functor sortToSet HSET := projSortToC sort Hsort HSET. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction_alt.v | projSortToSet | 38,256 |
Definition hat_functorSet : sort → HSET ⟶ sortToSet := hat_functor sort Hsort HSET CoproductsHSET. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction_alt.v | hat_functorSet | 38,257 |
Definition sorted_option_functorSet : sort → sortToSet ⟶ sortToSet := sorted_option_functor _ Hsort HSET TerminalHSET BinCoproductsHSET CoproductsHSET. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction_alt.v | sorted_option_functorSet | 38,258 |
Definition MultiSortedSigToSignatureSet : MultiSortedSig sort → Signature sortToSet sortToSet sortToSet. Proof. use MultiSortedSigToSignature. - apply TerminalHSET. - apply BinProductsHSET. - apply BinCoproductsHSET. - apply CoproductsHSET. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction_alt.v | MultiSortedSigToSignatureSet | 38,259 |
Definition MultiSortedSigToMonadSet (ms : MultiSortedSig sort) : Monad sortToSet. Proof. use MultiSortedSigToMonad. - apply TerminalHSET. - apply InitialHSET. - apply BinProductsHSET. - apply BinCoproductsHSET. - intros. apply ProductsHSET. - apply CoproductsHSET. - apply Exponentials_functor_HSET. - apply ColimsHSET_of_shape. - apply ms. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\MultiSortedMonadConstruction_alt.v | MultiSortedSigToMonadSet | 38,260 |
Definition PCF_Signature : Signature typeToSet _ _ := MultiSortedSigToSignatureSet type Htype PCF_Sig. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.StandardFiniteSets. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.STLC_alt. | SubstitutionSystems\SimplifiedHSS\PCF_alt.v | PCF_Signature | 38,261 |
Definition PCF_Functor : functor typeToSet2 typeToSet2 := Id_H _ BCtypeToSet PCF_Signature. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.StandardFiniteSets. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.STLC_alt. | SubstitutionSystems\SimplifiedHSS\PCF_alt.v | PCF_Functor | 38,262 |
Lemma PCF_Functor_Initial : Initial (FunctorAlg PCF_Functor). Proof. apply SignatureInitialAlgebra. - apply InitialHSET. - apply ColimsHSET_of_shape. - apply is_omega_cocont_MultiSortedSigToSignature. + intros; apply ProductsHSET. + apply Exponentials_functor_HSET. + apply ColimsHSET_of_shape. Defined. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.StandardFiniteSets. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.STLC_alt. | SubstitutionSystems\SimplifiedHSS\PCF_alt.v | PCF_Functor_Initial | 38,263 |
Definition PCF_Monad : Monad typeToSet := MultiSortedSigToMonadSet type Htype PCF_Sig. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.StandardFiniteSets. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.STLC_alt. | SubstitutionSystems\SimplifiedHSS\PCF_alt.v | PCF_Monad | 38,264 |
Definition PCF_M : typeToSet2 := alg_carrier _ (InitialObject PCF_Functor_Initial). | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.StandardFiniteSets. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.STLC_alt. | SubstitutionSystems\SimplifiedHSS\PCF_alt.v | PCF_M | 38,265 |
Definition var_map : typeToSet2⟦Id,PCF_M⟧ := η PCF_M_alg. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.StandardFiniteSets. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.STLC_alt. | SubstitutionSystems\SimplifiedHSS\PCF_alt.v | var_map | 38,266 |
Definition STLC_Sig : MultiSortedSig sort. Proof. use make_MultiSortedSig. - apply ((sort × sort) + (sort × sort))%set. - intros H; induction H as [st|st]; induction st as [s t]. + exact ((([],,arr s t) :: ([],,s) :: nil),,t). + exact (((cons s [],,t) :: []),,arr s t). Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | STLC_Sig | 38,267 |
Definition STLC_Signature : Signature (HSET / sort) _ _:= MultiSortedSigToSignature sort STLC_Sig. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | STLC_Signature | 38,268 |
Definition STLC_Functor : functor HSET_over_sort2 HSET_over_sort2 := Id_H STLC_Signature. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | STLC_Functor | 38,269 |
Lemma STLC_Functor_Initial : Initial (FunctorAlg STLC_Functor). Proof. apply SignatureInitialAlgebraSetSort. apply is_omega_cocont_MultiSortedSigToSignature. apply slice_precat_colims_of_shape, ColimsHSET_of_shape. Defined. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | STLC_Functor_Initial | 38,270 |
Definition STLC_Monad : Monad (HSET / sort) := MultiSortedSigToMonad sort STLC_Sig. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | STLC_Monad | 38,271 |
Definition STLC : HSET_over_sort2 := alg_carrier _ (InitialObject STLC_Functor_Initial). Let STLC_mor : HSET_over_sort2⟦STLC_Functor STLC,STLC⟧ := alg_map _ (InitialObject STLC_Functor_Initial). Let STLC_alg : algebra_ob STLC_Functor := InitialObject STLC_Functor_Initial. Local Lemma BP : BinProducts [HSET_over_sort,HSET]. Proof. apply BinProducts_functor_precat, BinProductsHSET. Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | STLC | 38,272 |
Definition var_map : HSET_over_sort2⟦1,STLC⟧ := SubstitutionSystems. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | var_map | 38,273 |
Definition app_source (s t : sort) (X : HSET_over_sort2) : HSET_over_sort2 := ((X ∙ proj_functor sort (arr s t)) ⊗ (X ∙ proj_functor sort s)) ∙ hat_functor sort t. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | app_source | 38,274 |
Definition app_map (s t : sort) : HSET_over_sort2⟦app_source s t STLC,STLC⟧ := (CoproductIn _ _ (Coproducts_functor_precat _ _ _ _ _) (ii1 (s,, t))) · SubstitutionSystems. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | app_map | 38,275 |
Definition lam_source (s t : sort) (X : HSET_over_sort2) : HSET_over_sort2 := (sorted_option_functor sort s ∙ X ∙ proj_functor sort t) ∙ hat_functor sort (arr s t). | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | lam_source | 38,276 |
Definition lam_map (s t : sort) : HSET_over_sort2⟦lam_source s t STLC,STLC⟧ := (CoproductIn _ _ (Coproducts_functor_precat _ _ _ _ _) (ii2 (s,,t))) · SubstitutionSystems. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | lam_map | 38,277 |
Definition make_STLC_Algebra X (fvar : HSET_over_sort2⟦1,X⟧) (fapp : ∏ s t, HSET_over_sort2⟦app_source s t X,X⟧) (flam : ∏ s t, HSET_over_sort2⟦lam_source s t X,X⟧) : algebra_ob STLC_Functor. Proof. apply (tpair _ X). use (BinCoproductArrow _ fvar). use CoproductArrow. intro b; induction b as [st|st]; induction st as [s t]. - apply (fapp s t). - apply (flam s t). Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | make_STLC_Algebra | 38,278 |
Definition foldr_map X (fvar : HSET_over_sort2⟦1,X⟧) (fapp : ∏ s t, HSET_over_sort2⟦app_source s t X,X⟧) (flam : ∏ s t, HSET_over_sort2⟦lam_source s t X,X⟧) : algebra_mor _ STLC_alg (make_STLC_Algebra X fvar fapp flam). Proof. apply (InitialArrow STLC_Functor_Initial (make_STLC_Algebra X fvar fapp flam)). Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | foldr_map | 38,279 |
Lemma foldr_var X (fvar : HSET_over_sort2⟦1,X⟧) (fapp : ∏ s t, HSET_over_sort2⟦app_source s t X,X⟧) (flam : ∏ s t, HSET_over_sort2⟦lam_source s t X,X⟧) : var_map · foldr_map X fvar fapp flam = fvar. Proof. assert (F := maponpaths (λ x, BinCoproductIn1 (BinCoproducts_functor_precat _ _ _ _ _) · x) (algebra_mor_commutes _ _ _ (foldr_map X fvar fapp flam))). rewrite assoc in F. eapply pathscomp0; [apply F|]. rewrite assoc. eapply pathscomp0; [eapply cancel_postcomposition, BinCoproductOfArrowsIn1|]. rewrite <- assoc. eapply pathscomp0; [eapply maponpaths, BinCoproductIn1Commutes|]. apply id_left. Defined. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Slice. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted. Require Import UniMath.SubstitutionSystems.MultiSorted. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction. | SubstitutionSystems\SimplifiedHSS\STLC.v | foldr_var | 38,280 |
Definition STLC_Signature : Signature sortToSet _ _ := MultiSortedSigToSignatureSet sort Hsort STLC_Sig. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | STLC_Signature | 38,281 |
Definition STLC_Functor : functor sortToSet2 sortToSet2 := Id_H _ BCsortToSet STLC_Signature. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | STLC_Functor | 38,282 |
Lemma STLC_Functor_Initial : Initial (FunctorAlg STLC_Functor). Proof. apply SignatureInitialAlgebra. - apply InitialHSET. - apply ColimsHSET_of_shape. - apply is_omega_cocont_MultiSortedSigToSignature. + intros; apply ProductsHSET. + apply Exponentials_functor_HSET. + apply ColimsHSET_of_shape. Defined. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | STLC_Functor_Initial | 38,283 |
Definition STLC_Monad : Monad sortToSet := MultiSortedSigToMonadSet sort Hsort STLC_Sig. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | STLC_Monad | 38,284 |
Definition STLC_M : sortToSet2 := alg_carrier _ (InitialObject STLC_Functor_Initial). | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | STLC_M | 38,285 |
Lemma STLC_Monad_ok : STLC_M = pr1 STLC_Monad. Proof. apply idpath. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | STLC_Monad_ok | 38,286 |
Definition var_map : sortToSet2⟦Id,STLC_M⟧ := SubstitutionSystems. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | var_map | 38,287 |
Definition app_source (s t : sort) : functor sortToSet2 sortToSet2 := (post_comp_functor (projSortToSet (s ⇒ t)) ⊗ post_comp_functor (projSortToSet s)) ∙ (post_comp_functor (hat_functorSet t)). | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | app_source | 38,288 |
Definition app_map (s t : sort) : sortToSet2⟦app_source s t STLC_M,STLC_M⟧ := CoproductIn _ _ (Coproducts_functor_precat _ _ _ _ (λ _, _)) (ii1 (s,,t)) · SubstitutionSystems. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | app_map | 38,289 |
Definition lam_source (s t : sort) : functor sortToSet2 sortToSet2 := pre_comp_functor (sorted_option_functorSet s) ∙ post_comp_functor (projSortToSet t) ∙ post_comp_functor (hat_functorSet (s ⇒ t)). | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | lam_source | 38,290 |
Definition lam_map (s t : sort) : sortToSet2⟦lam_source s t STLC_M,STLC_M⟧ := CoproductIn _ _ (Coproducts_functor_precat _ _ _ _ (λ _, _)) (ii2 (s,,t)) · SubstitutionSystems. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | lam_map | 38,291 |
Definition make_STLC_M_Algebra X (fvar : sortToSet2⟦Id,X⟧) (fapp : ∏ s t, sortToSet2⟦app_source s t X,X⟧) (flam : ∏ s t, sortToSet2⟦lam_source s t X,X⟧) : algebra_ob STLC_Functor. Proof. apply (tpair _ X), (BinCoproductArrow _ fvar), CoproductArrow; intros b. induction b as [st|st]; induction st as [s t]. - exact (fapp s t). - exact (flam s t). Defined. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | make_STLC_M_Algebra | 38,292 |
Definition foldr_map X (fvar : sortToSet2⟦Id,X⟧) (fapp : ∏ s t, sortToSet2⟦app_source s t X,X⟧) (flam : ∏ s t, sortToSet2⟦lam_source s t X,X⟧) : algebra_mor _ STLC_M_alg (make_STLC_M_Algebra X fvar fapp flam) := InitialArrow STLC_Functor_Initial (make_STLC_M_Algebra X fvar fapp flam). | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | foldr_map | 38,293 |
Lemma foldr_var X (fvar : sortToSet2⟦Id,X⟧) (fapp : ∏ s t, sortToSet2⟦app_source s t X,X⟧) (flam : ∏ s t, sortToSet2⟦lam_source s t X,X⟧) : var_map · foldr_map X fvar fapp flam = fvar. Proof. unfold var_map. unfold η, SubstitutionSystems.eta_from_alg, tau1_from_alg. rewrite <- assoc, (algebra_mor_commutes _ _ _ (foldr_map _ _ _ _)), assoc. etrans; [eapply cancel_postcomposition, BinCoproductOfArrowsIn1|]. rewrite id_left. apply BinCoproductIn1Commutes. Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | foldr_var | 38,294 |
Lemma foldr_app X (fvar : sortToSet2⟦Id,X⟧) (fapp : ∏ s t, sortToSet2⟦app_source s t X,X⟧) (flam : ∏ s t, sortToSet2⟦lam_source s t X,X⟧) (s t : sort) : app_map s t · foldr_map X fvar fapp flam = # (pr1 (app_source s t)) (foldr_map X fvar fapp flam) · fapp s t. Proof. etrans; [ apply cancel_postcomposition; unfold app_map; apply assoc |]. rewrite <- assoc. etrans; [apply maponpaths, (algebra_mor_commutes _ _ _ (foldr_map X fvar fapp flam))|]. rewrite assoc. etrans; [eapply cancel_postcomposition; rewrite <- assoc; apply maponpaths, BinCoproductOfArrowsIn2|]. rewrite <- !assoc. etrans; [apply maponpaths, maponpaths, BinCoproductIn2Commutes|]. rewrite assoc. etrans; [apply cancel_postcomposition; use (CoproductOfArrowsIn _ _ (Coproducts_functor_precat _ _ _ _ (λ _, _)))|]. rewrite <- assoc. apply maponpaths. exact (CoproductInCommutes _ _ _ _ _ _ (inl (s,,t))). Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | foldr_app | 38,295 |
Lemma foldr_lam X (fvar : sortToSet2⟦Id,X⟧) (fapp : ∏ s t, sortToSet2⟦app_source s t X,X⟧) (flam : ∏ s t, sortToSet2⟦lam_source s t X,X⟧) (s t : sort) : lam_map s t · foldr_map X fvar fapp flam = # (pr1 (lam_source s t)) (foldr_map X fvar fapp flam) · flam s t. Proof. etrans; [ apply cancel_postcomposition; unfold lam_map; apply assoc |]. rewrite <- assoc. etrans; [apply maponpaths, (algebra_mor_commutes _ _ _ (foldr_map X fvar fapp flam))|]. rewrite assoc. etrans; [eapply cancel_postcomposition; rewrite <- assoc; apply maponpaths, BinCoproductOfArrowsIn2|]. rewrite <- !assoc. etrans; [apply maponpaths, maponpaths, BinCoproductIn2Commutes|]. rewrite assoc. etrans; [apply cancel_postcomposition; use (CoproductOfArrowsIn _ _ (Coproducts_functor_precat _ _ _ _ (λ _, _)))|]. rewrite <- assoc. apply maponpaths. exact (CoproductInCommutes _ _ _ _ _ _ (inr (s,,t))). Qed. | Lemma | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | foldr_lam | 38,296 |
Definition psubst {X Y : sortToSet} (f : sortToSet⟦X, STLC Y ⟧) : sortToSet⟦ STLC (X ⊕ Y), STLC Y ⟧ := monadSubstGen_instantiated _ _ _ _ f. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | psubst | 38,297 |
Definition subst {X : sortToSet} (f : sortToSet⟦ 1, STLC X ⟧) : sortToSet⟦ STLC (1 ⊕ X), STLC X ⟧ := monadSubstGen_instantiated _ _ _ _ f. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | subst | 38,298 |
Definition weak {X Y : sortToSet} : sortToSet⟦STLC Y,STLC (X ⊕ Y)⟧ := mweak_instantiated sort Hsort HSET BinCoproductsHSET. | Definition | SubstitutionSystems | Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.MultiSortedBindingSig. Require Import UniMath.SubstitutionSystems.MultiSorted_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MultiSortedMonadConstruction_alt. Require Import UniMath.SubstitutionSystems.MonadsMultiSorted_alt. | SubstitutionSystems\SimplifiedHSS\STLC_alt.v | weak | 38,299 |
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