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Coq-UniMath

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Definition exch {X Y Z : sortToSet} : sortToSet⟦STLC (X ⊕ (Y ⊕ Z)), STLC (Y ⊕ (X ⊕ Z))⟧ := mexch_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

exch

Lemma psubst_interchange {X Y Z : sortToSet} (f : sortToSet⟦X,STLC (Y ⊕ Z)⟧) (g : sortToSet⟦Y, STLC Z⟧) : psubst f · psubst g = exch · psubst (g · weak) · psubst (f · psubst g). Proof. apply subst_interchange_law_gen_instantiated. 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

psubst_interchange

Lemma subst_interchange {X : sortToSet} (f : sortToSet⟦1,STLC (1 ⊕ X)⟧) (g : sortToSet⟦1,STLC X⟧) : subst f · subst g = exch · subst (g · weak) · subst (f · subst g). Proof. apply subst_interchange_law_gen_instantiated. 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

subst_interchange

Definition Const_plus_H (X : EndC) : functor EndC EndC := BinCoproduct_of_functors _ _ CPEndC (constant_functor _ _ X) H.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

Const_plus_H

Definition Id_H : functor EndC EndC := Const_plus_H (functor_identity _ : EndC).

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

Id_H

Definition eta_from_alg {X : EndC} (T : algebra_ob (Const_plus_H X)) : EndC ⟦ X , `T ⟧. Proof. exact (tau1_from_alg CPEndC (constant_functor _ _ X) H T). Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

eta_from_alg

Definition ptd_from_alg (T : algebra_ob Id_H) : Ptd. Proof. exists (pr1 T). exact (η T). Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

ptd_from_alg

Definition tau_from_alg {X : EndC} (T : algebra_ob (Const_plus_H X)) : EndC ⟦H `T, `T⟧. Proof. exact (tau2_from_alg CPEndC (constant_functor _ _ X) H T). Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

tau_from_alg

Definition bracket_property (T : algebra_ob Id_H) (f : U Z --> `T) (h : `T • (U Z) --> `T) : UU := alg_map _ T •• (U Z) · h = (identity (U Z) ⊕ θ `T) · (identity (U Z) ⊕ #H h) · (BinCoproductArrow (CPEndC _ _ ) f (tau_from_alg T)).

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

bracket_property

Definition bracket_at (T : algebra_ob Id_H) (f : U Z --> `T): UU := ∃! h : `T • (U Z) --> `T, bracket_property T f h.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

bracket_at

Definition bracket_property_parts (T : algebra_ob Id_H) (f : U Z --> `T) (h : `T • (U Z) --> `T) : UU := (f = η T •• (U Z) · h) × (θ `T · #H h · τ T = τ T •• (U Z) · h).

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

bracket_property_parts

Definition bracket_parts_at (T : algebra_ob Id_H) (f : U Z --> `T) : UU := ∃! h : `T • (U Z) --> `T, bracket_property_parts T f h.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

bracket_parts_at

Lemma parts_from_whole (T : algebra_ob Id_H) (f : U Z --> `T) (h : `T • (U Z) --> `T) : bracket_property T f h → bracket_property_parts T f h. Proof. intro Hyp. split. + unfold eta_from_alg. apply nat_trans_eq_alt. intro c. simpl. unfold coproduct_nat_trans_in1_data. assert (Hyp_inst := nat_trans_eq_pointwise Hyp c); clear Hyp. apply (maponpaths (λ m, BinCoproductIn1 (CP _ _)· m)) in Hyp_inst. match goal with |[ H1 : _ = ?f |- _ = _ ] => intermediate_path (f) end. * clear Hyp_inst. rewrite <- assoc. apply BinCoproductIn1Commutes_right_in_ctx_dir. rewrite id_left. apply BinCoproductIn1Commutes_right_in_ctx_dir. rewrite id_left. apply BinCoproductIn1Commutes_right_dir. apply idpath. * rewrite <- Hyp_inst; clear Hyp_inst. rewrite <- assoc. apply idpath. + unfold tau_from_alg. apply nat_trans_eq_alt. intro c. simpl. unfold coproduct_nat_trans_in2_data. assert (Hyp_inst := nat_trans_eq_pointwise Hyp c); clear Hyp. apply (maponpaths (λ m, BinCoproductIn2 (CP _ _)· m)) in Hyp_inst. match goal with |[ H1 : _ = ?f |- _ = _ ] => intermediate_path (f) end. * clear Hyp_inst. do 2 rewrite <- assoc. apply BinCoproductIn2Commutes_right_in_ctx_dir. simpl. rewrite <- assoc. apply maponpaths. apply BinCoproductIn2Commutes_right_in_ctx_dir. simpl. rewrite <- assoc. apply maponpaths. unfold tau_from_alg. apply BinCoproductIn2Commutes_right_dir. apply idpath. * rewrite <- Hyp_inst; clear Hyp_inst. rewrite <- assoc. apply idpath. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

parts_from_whole

Lemma whole_from_parts (T : algebra_ob Id_H) (f : U Z --> `T) (h : `T • (U Z) --> `T) : bracket_property_parts T f h → bracket_property T f h. Proof. intros [Hyp1 Hyp2]. apply nat_trans_eq_alt. intro c. apply BinCoproductArrow_eq_cor. + clear Hyp2. assert (Hyp1_inst := nat_trans_eq_pointwise Hyp1 c); clear Hyp1. rewrite <- assoc. apply BinCoproductIn1Commutes_right_in_ctx_dir. rewrite id_left. apply BinCoproductIn1Commutes_right_in_ctx_dir. rewrite id_left. apply BinCoproductIn1Commutes_right_dir. simpl. simpl in Hyp1_inst. rewrite Hyp1_inst. simpl. apply assoc. + clear Hyp1. assert (Hyp2_inst := nat_trans_eq_pointwise Hyp2 c); clear Hyp2. rewrite <- assoc. apply BinCoproductIn2Commutes_right_in_ctx_dir. simpl. rewrite assoc. eapply pathscomp0. * eapply pathsinv0. exact Hyp2_inst. * clear Hyp2_inst. simpl. do 2 rewrite <- assoc. apply maponpaths. apply BinCoproductIn2Commutes_right_in_ctx_dir. simpl. rewrite <- assoc. apply maponpaths. apply BinCoproductIn2Commutes_right_dir. apply idpath. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

whole_from_parts

Definition bracket_property_parts_identity_nicer (h : `T • `T --> `T) : UU := (identity `T = η T •• `T · h) × (θ `T · #H h · τ T = τ T •• `T · h).

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

bracket_property_parts_identity_nicer

Lemma bracket_property_parts_identity_nicer_impl1 (h : `T • `T --> `T): bracket_property_parts θ T (identity _) h -> bracket_property_parts_identity_nicer h. Proof. intro Hyp. induction Hyp as [Hyp1 Hyp2]. split. - etrans. 2: { exact Hyp1. } apply nat_trans_eq_alt. intro c. apply idpath. - etrans. { exact Hyp2. } apply idpath. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

bracket_property_parts_identity_nicer_impl1

Lemma bracket_property_parts_identity_nicer_impl2 (h : `T • `T --> `T): bracket_property_parts_identity_nicer h -> bracket_property_parts θ T (identity _) h.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

bracket_property_parts_identity_nicer_impl2

Definition heterogeneous_substitution : UU := ∑ (T: algebra_ob Id_H), ∑ (θ : @PrestrengthForSignatureAtPoint C C C H (ptd_from_alg T)), bracket_at θ T (identity _).

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

heterogeneous_substitution

Definition θ_from_hetsubst (T : heterogeneous_substitution) : @PrestrengthForSignatureAtPoint C C C H (ptd_from_alg T) := pr1 (pr2 T).

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

θ_from_hetsubst

Definition prejoin_from_hetsubst (T : heterogeneous_substitution) : `T • `T --> `T := pr1 (pr1 (pr2 (pr2 T))).

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

prejoin_from_hetsubst

Lemma prejoin_from_hetsubst_η (T : heterogeneous_substitution) : identity `T = η T •• `T · (prejoin_from_hetsubst T). Proof. refine (pr1 (bracket_property_parts_identity_nicer_impl1 T (θ_from_hetsubst T) _ _)). apply parts_from_whole. exact (pr2 (pr1 (pr2 (pr2 T)))). Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

prejoin_from_hetsubst_η

Lemma prejoin_from_hetsubst_τ (T : heterogeneous_substitution) : θ_from_hetsubst T `T · #H (prejoin_from_hetsubst T) · τ T = τ T •• `T · (prejoin_from_hetsubst T). Proof. refine (pr2 (bracket_property_parts_identity_nicer_impl1 T (θ_from_hetsubst T) _ _)). apply parts_from_whole. exact (pr2 (pr1 (pr2 (pr2 T)))). Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

prejoin_from_hetsubst_τ

Definition bracket (T : algebra_ob Id_H) : UU := ∏ (Z : Ptd) (f : U Z --> `T), bracket_at (nat_trans_fix_snd_arg _ _ _ _ _ θ Z) T f.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

bracket

Lemma isaprop_bracket (T : algebra_ob Id_H) : isaprop (bracket T). Proof. apply impred_isaprop; intro Z. apply impred_isaprop; intro f. apply isapropiscontr. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

isaprop_bracket

Definition bracket_parts (T : algebra_ob Id_H) : UU := ∏ (Z : Ptd) (f : U Z --> `T), bracket_parts_at (nat_trans_fix_snd_arg _ _ _ _ _ θ Z) T f.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

bracket_parts

Definition hss : UU := ∑ (T: algebra_ob Id_H), bracket H θ T. Coercion hetsubst_from_hss (T : hss) : heterogeneous_substitution H. Proof. exists (pr1 T). use tpair. - apply (nat_trans_fix_snd_arg _ _ _ _ _ θ). - apply (pr2 T). Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

hss

Definition fbracket (T : hss) (Z : Ptd) (f : U Z --> `T) : `T • (U Z) --> `T := pr1 (pr1 (pr2 T Z f)).

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

fbracket

Definition fbracket_unique_pointwise (T : hss) {Z : Ptd} (f : U Z --> `T) : ∏ (α : functor_composite (U Z) `T ⟹ pr1 `T), (∏ c : C, pr1 f c = pr1 (η T) (pr1 (U Z) c) · α c) → (∏ c : C, pr1 (θ (`T ⊗ Z)) c · pr1 (#H α) c · pr1 (τ T) c = pr1 (τ T) (pr1 (U Z) c) · α c) → α = ⦃f⦄_{Z}.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

fbracket_unique_pointwise

Definition fbracket_unique (T : hss) {Z : Ptd} (f : U Z --> `T) : ∏ α : `T • (U Z) --> `T, bracket_property_parts H (nat_trans_fix_snd_arg _ _ _ _ _ θ Z) T f α → α = ⦃f⦄_{Z}.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

fbracket_unique

Definition fbracket_unique_target_pointwise (T : hss) {Z : Ptd} (f : U Z --> `T) : ∏ α : `T • U Z --> `T, bracket_property_parts H (nat_trans_fix_snd_arg _ _ _ _ _ θ Z) T f α → ∏ c, pr1 α c = pr1 ⦃f⦄_{Z} c.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

fbracket_unique_target_pointwise

Lemma fbracket_η (T : hss) : ∏ {Z : Ptd} (f : U Z --> `T), f = η T •• U Z · ⦃f⦄_{Z}. Proof. intros Z f. exact (pr1 (parts_from_whole _ _ _ _ _ (pr2 (pr1 (pr2 T Z f))))). Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

fbracket_η

Lemma fbracket_τ (T : hss) : ∏ {Z : Ptd} (f : U Z --> `T), θ (`T ⊗ Z) · #H ⦃f⦄_{Z} · τ T = τ T •• U Z · ⦃f⦄_{Z}. Proof. intros Z f. exact (pr2 (parts_from_whole _ _ _ _ _ (pr2 (pr1 (pr2 T Z f))))). Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

fbracket_τ

Lemma fbracket_natural (T : hss) {Z Z' : Ptd} (f : Z --> Z') (g : U Z' --> `T) : (`T ∘ #U f : EndC⟦ `T • U Z , `T • U Z' ⟧) · ⦃g⦄_{Z'} = ⦃#U f · g⦄_{Z}. Proof. apply fbracket_unique_pointwise. - simpl. intro c. rewrite assoc. pose proof (nat_trans_ax (η T)) as H'. simpl in H'. rewrite <- H'; clear H'. rewrite <- assoc. apply maponpaths. pose proof (nat_trans_eq_weq (homset_property C) _ _ (fbracket_η T g)) as X. simpl in X. exact (X _ ). - intro c; simpl. assert (H':=nat_trans_ax (τ T)). simpl in H'. eapply pathscomp0. 2: apply assoc'. eapply pathscomp0. 2: { apply cancel_postcomposition. apply H'. } clear H'. set (H':=fbracket_τ T g). simpl in H'. assert (X:= nat_trans_eq_pointwise H' c). simpl in X. rewrite <- assoc. rewrite <- assoc. transitivity ( # (pr1 (H ((`T)))) (pr1 f c) · (pr1 (θ ((`T) ⊗ Z')) c)· pr1 (# H ⦃g⦄_{Z'}) c· pr1 (τ T) c). 2: { rewrite <- assoc. rewrite <- assoc. apply maponpaths. repeat rewrite assoc. apply X. } clear X. set (A:=θ_nat_2_pointwise). simpl in *. set (A':= A C C C H θ (`T) Z Z'). simpl in A'. set (A2:= A' f). clearbody A2; clear A'; clear A. rewrite A2; clear A2. repeat rewrite <- assoc. apply maponpaths. simpl. repeat rewrite assoc. apply cancel_postcomposition. rewrite (functor_comp H). apply cancel_postcomposition. clear H'. set (A:=horcomp_id_postwhisker C C C). rewrite A; try apply homset_property. apply idpath. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

fbracket_natural

Lemma compute_fbracket (T : hss) : ∏ {Z : Ptd} (f : Z --> ptd_from_alg T), ⦃#U f⦄_{Z} = (`T ∘ # U f : EndC⟦`T • U Z , `T • U (ptd_from_alg T)⟧) · ⦃identity (U (ptd_from_alg T))⦄_{ptd_from_alg T}. Proof. intros Z f. assert (A : f = f · identity _ ). { rewrite id_right; apply idpath. } rewrite A. rewrite functor_comp. rewrite <- fbracket_natural. rewrite id_right. apply idpath. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

compute_fbracket

Lemma heterogeneous_substitution_into_bracket {Z : Ptd} (f : Z --> ptd_from_alg T0) : bracket_property H (nat_trans_fix_snd_arg _ _ _ _ _ θ Z) T0 (#U f) ((` T0 ∘ #U f : EndC ⟦ `T0 • U Z , `T0 • U (ptd_from_alg T0) ⟧) · prejoin_from_hetsubst T0). Proof. apply whole_from_parts. split. - apply nat_trans_eq_alt. intro c. induction f as [f pt]. simpl. assert (alg_map_nat := nat_trans_ax (alg_map Id_H T0) _ _ (pr1 f c)). etrans. 2: { rewrite <- assoc. apply maponpaths. rewrite assoc. apply cancel_postcomposition. exact alg_map_nat. } clear alg_map_nat. etrans. 2: { do 2 rewrite assoc. do 2 apply cancel_postcomposition. apply pathsinv0. unfold Id_H. simpl. apply BinCoproductIn1Commutes. } simpl. etrans. { apply pathsinv0. apply id_right. } do 2 rewrite <- assoc. apply maponpaths. rewrite assoc. assert (prejoin_ok := prejoin_from_hetsubst_η T0). apply (maponpaths pr1) in prejoin_ok. apply toforallpaths in prejoin_ok. apply prejoin_ok. - rewrite functor_comp. apply nat_trans_eq_alt. intro c. induction f as [f pt]. simpl. assert (alg_map_nat := nat_trans_ax (alg_map Id_H T0) _ _ (pr1 f c)). etrans. 2: { rewrite <- assoc. apply maponpaths. rewrite assoc. apply cancel_postcomposition. exact alg_map_nat. } etrans. 2: { do 2 rewrite assoc. do 2 apply cancel_postcomposition. apply pathsinv0. unfold Id_H. simpl. apply BinCoproductIn2Commutes. } assert (prejoin_ok := prejoin_from_hetsubst_τ T0). apply (maponpaths pr1) in prejoin_ok. apply toforallpaths in prejoin_ok. assert (prejoin_ok_inst := prejoin_ok c). simpl in prejoin_ok_inst. etrans. { repeat rewrite assoc. do 3 apply cancel_postcomposition. apply pathsinv0. assert (theta_nat_2 := θ_nat_2_pointwise _ _ _ H θ `T0 _ _ (f,,pt) c). rewrite horcomp_id_postwhisker in theta_nat_2; try apply homset_property. apply theta_nat_2. } etrans. { repeat rewrite <- assoc. apply maponpaths. rewrite assoc. exact prejoin_ok_inst. } clear prejoin_ok prejoin_ok_inst. repeat rewrite assoc. apply idpath. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

heterogeneous_substitution_into_bracket

Definition isAlgMor {T T' : Alg} (f : T --> T') : UU := #H (# U f) · τ T' = compose (C:=EndC) (τ T) (#U f).

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

isAlgMor

Lemma isaprop_isAlgMor (T T' : Alg) (f : T --> T') : isaprop (isAlgMor f). Proof. apply isaset_nat_trans. apply hs. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

isaprop_isAlgMor

Lemma τ_part_of_alg_mor (T T' : @algebra_ob [C, C] Id_H) (β : @algebra_mor [C, C] Id_H T T'): #H β · τ T' = compose (C:=EndC) (τ T) β. Proof. assert (β_is_alg_mor := pr2 β). simpl in β_is_alg_mor. assert (β_is_alg_mor_inst := maponpaths (fun m:EndC⟦_,_⟧ => (BinCoproductIn2 (CPEndC _ _))· m) β_is_alg_mor); clear β_is_alg_mor. simpl in β_is_alg_mor_inst. apply nat_trans_eq_alt. intro c. assert (β_is_alg_mor_inst':= nat_trans_eq_pointwise β_is_alg_mor_inst c); clear β_is_alg_mor_inst. simpl in β_is_alg_mor_inst'. rewrite assoc in β_is_alg_mor_inst'. eapply pathscomp0. 2: { eapply pathsinv0. exact β_is_alg_mor_inst'. } clear β_is_alg_mor_inst'. apply BinCoproductIn2Commutes_right_in_ctx_dir. simpl. rewrite <- assoc. apply idpath. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

τ_part_of_alg_mor

Lemma is_ptd_mor_alg_mor (T T' : @algebra_ob [C, C] Id_H) (β : @algebra_mor [C, C] Id_H T T') : @is_ptd_mor C (ptd_from_alg T) (ptd_from_alg T') (pr1 β). Proof. simpl. unfold is_ptd_mor. simpl. intro c. rewrite <- assoc. assert (X:=pr2 β). assert (X':= nat_trans_eq_pointwise X c). simpl in *. etrans. { apply maponpaths. apply X'. } unfold coproduct_nat_trans_in1_data. repeat rewrite assoc. unfold coproduct_nat_trans_data. etrans. { apply cancel_postcomposition. apply BinCoproductIn1Commutes. } simpl. repeat rewrite <- assoc. apply id_left. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

is_ptd_mor_alg_mor

Definition ptd_from_alg_mor {T T' : algebra_ob Id_H} (β : algebra_mor _ T T') : ptd_from_alg T --> ptd_from_alg T'. Proof. exists (pr1 β). apply is_ptd_mor_alg_mor. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

ptd_from_alg_mor

Definition ptd_from_alg_functor_data : functor_data (category_FunctorAlg Id_H) Ptd. Proof. exists ptd_from_alg. intros T T' β. apply ptd_from_alg_mor. exact β. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

ptd_from_alg_functor_data

Lemma is_functor_ptd_from_alg_functor_data : is_functor ptd_from_alg_functor_data. Proof. split; simpl; intros. + unfold functor_idax. intro T. apply (invmap (eq_ptd_mor_cat _ _ _)). apply (invmap (eq_ptd_mor _ _)). apply idpath. + unfold functor_compax. intros T T' T'' β β'. apply (invmap (eq_ptd_mor_cat _ _ _)). apply (invmap (eq_ptd_mor _ _)). apply idpath. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

is_functor_ptd_from_alg_functor_data

Definition ptd_from_alg_functor: functor (category_FunctorAlg Id_H) Ptd := tpair _ _ is_functor_ptd_from_alg_functor_data.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

ptd_from_alg_functor

Definition isbracketMor {T T' : hss} (β : algebra_mor _ T T') : UU := ∏ (Z : Ptd) (f : U Z --> `T), ⦃f⦄_{Z} · β = β •• U Z · ⦃f · #U (# ptd_from_alg_functor β)⦄_{Z}.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

isbracketMor

Lemma isaprop_isbracketMor (T T' : hss) (β : algebra_mor _ T T') : isaprop (isbracketMor β). Proof. do 2 (apply impred; intro). apply isaset_nat_trans. apply homset_property. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

isaprop_isbracketMor

Definition ishssMor {T T' : hss} (β : algebra_mor _ T T') : UU := isbracketMor β.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

ishssMor

Definition hssMor (T T' : hss) : UU := ∑ β : algebra_mor _ T T', ishssMor β.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

hssMor

Definition isAlgMor_hssMor {T T' : hss} (β : hssMor T T') : isAlgMor β := pr1 (pr2 β).

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

isAlgMor_hssMor

Definition isbracketMor_hssMor {T T' : hss} (β : hssMor T T') : isbracketMor β := pr2 β.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

isbracketMor_hssMor

Definition hssMor_eq1 : β = β' ≃ (pr1 β = pr1 β'). Proof. apply subtypeInjectivity. intro γ. apply isaprop_isbracketMor. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

hssMor_eq1

Definition hssMor_eq : β = β' ≃ (β : EndC ⟦ _ , _ ⟧) = β'. Proof. eapply weqcomp. - apply hssMor_eq1. - apply subtypeInjectivity. intro. apply isaset_nat_trans. apply homset_property. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

hssMor_eq

Lemma isaset_hssMor (T T' : hss) : isaset (hssMor T T'). Proof. intros β β'. apply (isofhlevelweqb _ (hssMor_eq _ _ β β')). apply isaset_nat_trans. apply homset_property. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

isaset_hssMor

Lemma ishssMor_id (T : hss) : ishssMor (identity (C:=category_FunctorAlg _) (pr1 T)). Proof. unfold ishssMor. unfold isbracketMor. intros Z f. rewrite id_right. rewrite functor_id. rewrite id_right. apply pathsinv0. set (H2:=pre_composition_functor _ _ C (U Z)). set (H2' := functor_id H2). simpl in H2'. simpl. rewrite H2'. rewrite (@id_left EndC). apply idpath. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

ishssMor_id

Definition hssMor_id (T : hss) : hssMor _ _ := tpair _ _ (ishssMor_id T).

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

hssMor_id

Lemma ishssMor_comp {T T' T'' : hss} (β : hssMor T T') (γ : hssMor T' T'') : ishssMor (compose (C:=category_FunctorAlg _) (pr1 β) (pr1 γ)). Proof. unfold ishssMor. unfold isbracketMor. intros Z f. eapply pathscomp0; [apply assoc|]. etrans. { apply cancel_postcomposition. apply isbracketMor_hssMor. } rewrite <- assoc. etrans. { apply maponpaths. apply isbracketMor_hssMor. } rewrite assoc. do 2 rewrite functor_comp. rewrite assoc. apply cancel_postcomposition. apply pathsinv0, (functor_comp (pre_composition_functor _ _ C (U Z)) ). Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

ishssMor_comp

Definition hssMor_comp {T T' T'' : hss} (β : hssMor T T') (γ : hssMor T' T'') : hssMor T T'' := tpair _ _ (ishssMor_comp β γ).

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

hssMor_comp

Definition hss_obmor : precategory_ob_mor. Proof. exists hss. exact hssMor. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

hss_obmor

Definition hss_precategory_data : precategory_data. Proof. exists hss_obmor. split. - exact hssMor_id. - exact @hssMor_comp. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

hss_precategory_data

Lemma is_precategory_hss : is_precategory hss_precategory_data. Proof. apply is_precategory_one_assoc_to_two. repeat split; intros. - apply (invmap (hssMor_eq _ _ _ _ )). apply (@id_left EndC). - apply (invmap (hssMor_eq _ _ _ _ )). apply (@id_right EndC). - apply (invmap (hssMor_eq _ _ _ _ )). apply (@assoc EndC). Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

is_precategory_hss

Definition hss_precategory : precategory := tpair _ _ is_precategory_hss.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

hss_precategory

Lemma has_homsets_precategory_hss : has_homsets hss_precategory. Proof. red. intros T T'. apply isaset_hssMor. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

has_homsets_precategory_hss

Definition hss_category : category := hss_precategory ,, has_homsets_precategory_hss.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems.v

hss_category

Definition GenMendlerIteration : ∏ (C : category) (F : functor C C) (μF_Initial : Initial (FunctorAlg F)) (C' : category) (X : C') (L : functor C C'), is_left_adjoint L → ∏ ψ : ψ_source C C' X L ⟹ ψ_target C F C' X L, ∃! h : C' ⟦ L ` (InitialObject μF_Initial), X ⟧, # L (alg_map F (InitialObject μF_Initial)) · h = ψ ` (InitialObject μF_Initial) h. Proof. simpl. apply GenMendlerIteration. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

GenMendlerIteration

Theorem fusion_law : ∏ (C : category) (F : functor C C) (μF_Initial : Initial (category_FunctorAlg F)) (C' : category) (X X' : C') (L : functor C C') (is_left_adj_L : is_left_adjoint L) (ψ : ψ_source C C' X L ⟹ ψ_target C F C' X L) (L' : functor C C') (is_left_adj_L' : is_left_adjoint L') (ψ' : ψ_source C C' X' L' ⟹ ψ_target C F C' X' L') (Φ : yoneda_objects C' X • functor_opp L ⟹ yoneda_objects C' X' • functor_opp L'), let T:= (` (InitialObject μF_Initial)) in ψ T · Φ (F T) = Φ T · ψ' T → Φ T (It μF_Initial X L is_left_adj_L ψ) = It μF_Initial X' L' is_left_adj_L' ψ'. Proof. apply fusion_law. Qed.

Theorem

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

fusion_law

Lemma fbracket_natural : ∏ (C : category) (CP : BinCoproducts C) (H : Presignature C C C) (T : hss CP H) (Z Z' : category_Ptd C) (f : category_Ptd C ⟦ Z, Z' ⟧) (g : [C,C] ⟦ U Z', `T ⟧), (`T ∘ # U f : [C, C] ⟦ `T • U Z , `T • U Z' ⟧) · ⦃g⦄_{Z'} = ⦃#U f · g⦄_{Z} . Proof. apply fbracket_natural. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

fbracket_natural

Lemma compute_fbracket : ∏ (C : category) (CP : BinCoproducts C) (H : Presignature C C C) (T : hss CP H) (Z : category_Ptd C) (f : category_Ptd C ⟦ Z, ptd_from_alg T ⟧), ⦃#U f⦄_{Z} = (`T ∘ # U f : [C, C] ⟦ `T • U Z , `T • U _ ⟧) · ⦃ identity (U (ptd_from_alg T)) ⦄_{ptd_from_alg T}. Proof. apply compute_fbracket. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

compute_fbracket

Definition Monad_from_hss : ∏ (C : category) (CP : BinCoproducts C) (H : Signature C C C), hss CP H → Monad C. Proof. apply Monad_from_hss. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

Monad_from_hss

Definition hss_to_monad_functor : ∏ (C : category) (CP : BinCoproducts C) (H : Signature C C C), functor (hss_precategory CP H) (category_Monad C). Proof. apply hss_to_monad_functor. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

hss_to_monad_functor

Lemma faithful_hss_to_monad : ∏ (C : category) (CP : BinCoproducts C) (H : Signature C C C), faithful (hss_to_monad_functor C CP H). Proof. apply faithful_hss_to_monad. Defined.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

faithful_hss_to_monad

Definition bracket_for_initial_algebra : ∏ (C : category) (CP : BinCoproducts C), (∏ Z : category_Ptd C, GlobalRightKanExtensionExists C C (U Z) C) → ∏ (H : Presignature C C C) (IA : Initial (FunctorAlg (Id_H C CP H))) (Z : category_Ptd C), [C, C] ⟦ U Z, U (ptd_from_alg (InitAlg C CP H IA)) ⟧ → [C, C] ⟦ ℓ (U Z) ` (InitialObject IA), ` (InitAlg C CP H IA) ⟧. Proof. apply bracket_Thm15. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

bracket_for_initial_algebra

Lemma bracket_Thm15_ok_η : ∏ (C : category) (CP : BinCoproducts C) (KanExt : ∏ Z : category_Ptd C, GlobalRightKanExtensionExists C C (U Z) C) (H : Presignature C C C) (IA : Initial (FunctorAlg (Id_H C CP H))) (Z : category_Ptd C) (f : [C,C] ⟦ U Z, U (ptd_from_alg (InitAlg C CP H IA))⟧), f = # (pr1 (ℓ (U Z))) (η (InitAlg C CP H IA)) · bracket_Thm15 C CP KanExt H IA Z f. Proof. apply bracket_Thm15_ok_part1. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

bracket_Thm15_ok_η

Lemma bracket_Thm15_ok_τ : ∏ (C : category) (CP : BinCoproducts C) (KanExt : ∏ Z : category_Ptd C, GlobalRightKanExtensionExists C C (U Z) C) (H : Presignature C C C) (IA : Initial (FunctorAlg (Id_H C CP H))) (Z : category_Ptd C) (f : [C,C] ⟦ U Z, U (ptd_from_alg (InitAlg C CP H IA)) ⟧), (theta H) (` (InitAlg C CP H IA) ⊗ Z) · # H (bracket_Thm15 C CP KanExt H IA Z f) · τ (InitAlg C CP H IA) = # (pr1 (ℓ (U Z))) (τ (InitAlg C CP H IA)) · bracket_Thm15 C CP KanExt H IA Z f. Proof. apply bracket_Thm15_ok_part2. Qed.

Lemma

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

bracket_Thm15_ok_τ

Definition Initial_HSS : ∏ (C : category) (CP : BinCoproducts C), (∏ Z : category_Ptd C, GlobalRightKanExtensionExists C C (U Z) C) → ∏ H : Presignature C C C, Initial (FunctorAlg (Id_H C CP H)) → Initial (hss_category CP H). Proof. apply InitialHSS. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

Initial_HSS

Definition Sum_of_Signatures : ∏ (C D D': category), BinCoproducts D → Signature C D D' → Signature C D D' → Signature C D D'. Proof. apply BinSum_of_Signatures. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

Sum_of_Signatures

Definition App_Sig : ∏ (C : category), BinProducts C → Signature C C C. Proof. apply App_Sig. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

App_Sig

Definition Lam_Sig : ∏ (C : category), Terminal C → BinCoproducts C → BinProducts C → Signature C C C. Proof. apply Lam_Sig. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

Lam_Sig

Definition Flat_Sig : ∏ (C : category), Signature C C C. Proof. apply Flat_Sig. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

Flat_Sig

Definition Lam_Flatten : ∏ (C : category) (terminal : Terminal C) (CC : BinCoproducts C) (CP : BinProducts C), (∏ Z : category_Ptd C, GlobalRightKanExtensionExists C C (U Z) C) → ∏ Lam_Initial : Initial (FunctorAlg (Id_H C CC (Lam_Sig C terminal CC CP))), [C, C] ⟦ (Flat_H C) ` (InitialObject Lam_Initial), ` (InitialObject Lam_Initial) ⟧. Proof. apply Lam_Flatten. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

Lam_Flatten

Definition fbracket_for_LamE_algebra_on_Lam : ∏ (C : category) (terminal : Terminal C) (CC : BinCoproducts C) (CP : BinProducts C) (KanExt : ∏ Z : category_Ptd C, GlobalRightKanExtensionExists C C (U Z) C) (Lam_Initial : Initial (FunctorAlg (Id_H C CC (Lam_Sig C terminal CC CP)))) (Z : category_Ptd C), category_Ptd C ⟦ Z , (ptd_from_alg_functor CC (LamE_Sig C terminal CC CP)) (LamE_algebra_on_Lam C terminal CC CP KanExt Lam_Initial) ⟧ → [C, C] ⟦ functor_composite (U Z) ` (LamE_algebra_on_Lam C terminal CC CP KanExt Lam_Initial), ` (LamE_algebra_on_Lam C terminal CC CP KanExt Lam_Initial) ⟧. Proof. apply fbracket_for_LamE_algebra_on_Lam. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

fbracket_for_LamE_algebra_on_Lam

Definition EVAL : ∏ (C : category) (terminal : Terminal C) (CC : BinCoproducts C) (CP : BinProducts C) (KanExt : ∏ Z : category_Ptd C, GlobalRightKanExtensionExists C C (U Z) C) (Lam_Initial : Initial (FunctorAlg (Id_H C CC (LamSignature.Lam_Sig C terminal CC CP)))) (LamE_Initial : Initial (FunctorAlg (Id_H C CC (LamE_Sig C terminal CC CP)))), hss_category CC (LamE_Sig C terminal CC CP) ⟦ InitialObject (LamEHSS_Initial C terminal CC CP KanExt LamE_Initial), LamE_model_on_Lam C terminal CC CP KanExt Lam_Initial ⟧. Proof. apply FLATTEN. Defined.

Definition

SubstitutionSystems

Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.

SubstitutionSystems\SimplifiedHSS\SubstitutionSystems_Summary.v

EVAL

Definition top_disp_cat_ob_mor : disp_cat_ob_mor hset_category. Proof. use make_disp_cat_ob_mor. - intro X. exact (isTopologicalSpace X). - intros X Y T U f. exact (@continuous (X,,T) (Y,,U) f). Defined.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Require Import UniMath.Topology.Topology. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core.

Topology\CategoryTop.v

top_disp_cat_ob_mor

Definition top_disp_cat_data : disp_cat_data hset_category. Proof. exists top_disp_cat_ob_mor. split. - do 5 intro. assumption. - intros *. apply continuous_funcomp. Defined.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Require Import UniMath.Topology.Topology. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core.

Topology\CategoryTop.v

top_disp_cat_data

Definition top_disp_cat_axioms : disp_cat_axioms hset_category top_disp_cat_data. Proof. do 3 (split ; intros ; try (apply proofirrelevance, isaprop_continuous)). apply isasetaprop, isaprop_continuous. Defined.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Require Import UniMath.Topology.Topology. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core.

Topology\CategoryTop.v

top_disp_cat_axioms

Definition disp_top : disp_cat hset_category := _ ,, top_disp_cat_axioms.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Require Import UniMath.Topology.Topology. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core.

Topology\CategoryTop.v

disp_top

Definition isfilter_imply := ∏ A B : X → hProp, (∏ x : X, A x → B x) → F A → F B.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isfilter_imply

Lemma isaprop_isfilter_imply : isaprop isfilter_imply. Proof. apply impred_isaprop ; intro A. apply impred_isaprop ; intro B. apply isapropimpl. apply isapropimpl. apply propproperty. Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isaprop_isfilter_imply

Definition isfilter_finite_intersection := ∏ (L : Sequence (X → hProp)), (∏ n, F (L n)) → F (finite_intersection L).

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isfilter_finite_intersection

Lemma isaprop_isfilter_finite_intersection : isaprop isfilter_finite_intersection. Proof. apply impred_isaprop ; intros L. apply isapropimpl. apply propproperty. Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isaprop_isfilter_finite_intersection

Definition isfilter_htrue : hProp := F (λ _ : X, htrue).

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isfilter_htrue

Definition isfilter_and := ∏ A B : X → hProp, F A → F B → F (λ x : X, A x ∧ B x).

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isfilter_and

Lemma isaprop_isfilter_and : isaprop isfilter_and. Proof. apply impred_isaprop ; intro A. apply impred_isaprop ; intro B. apply isapropimpl. apply isapropimpl. apply propproperty. Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isaprop_isfilter_and

Definition isfilter_notempty := ∏ A : X → hProp, F A → ∃ x : X, A x.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isfilter_notempty

Lemma isaprop_isfilter_notempty : isaprop isfilter_notempty. Proof. apply impred_isaprop ; intro A. apply isapropimpl. apply propproperty. Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isaprop_isfilter_notempty

Lemma isfilter_finite_intersection_htrue : isfilter_finite_intersection → isfilter_htrue. Proof. intros Hand. unfold isfilter_htrue. rewrite <- finite_intersection_htrue. apply Hand. intros m. apply fromempty. generalize (pr2 m). now apply negnatlthn0. Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isfilter_finite_intersection_htrue

Lemma isfilter_finite_intersection_and : isfilter_finite_intersection → isfilter_and. Proof. intros Hand A B Fa Fb. rewrite <- finite_intersection_and. apply Hand. intros m. induction m as [m Hm]. induction m as [ | m _]. - exact Fa. - exact Fb. Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isfilter_finite_intersection_and

Lemma isfilter_finite_intersection_carac : isfilter_htrue → isfilter_and → isfilter_finite_intersection. Proof. intros Htrue Hand L. apply (pr2 (finite_intersection_hProp F)). split. - exact Htrue. - exact Hand. Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isfilter_finite_intersection_carac

Definition isPreFilter {X : UU} (F : (X → hProp) → hProp) := isfilter_imply F × isfilter_finite_intersection F.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

isPreFilter

Definition PreFilter (X : UU) := ∑ (F : (X → hProp) → hProp), isPreFilter F.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

PreFilter

Definition make_PreFilter {X : UU} (F : (X → hProp) → hProp) (Himpl : isfilter_imply F) (Htrue : isfilter_htrue F) (Hand : isfilter_and F) : PreFilter X := F,, Himpl,, isfilter_finite_intersection_carac F Htrue Hand.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

make_PreFilter

Definition pr1PreFilter (X : UU) (F : PreFilter X) : (X → hProp) → hProp := pr1 F.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

pr1PreFilter