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⌀
ptd_from_alg_functor: functor (category_FunctorAlg Id_H) Ptd := tpair _ _ is_functor_ptd_from_alg_functor_data.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
ptd_from_alg_functor
null
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
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
isbracketMor
null
isaprop_isbracketMor (T T' : hss) (β : algebra_mor _ T T') : isaprop (isbracketMor β).
Lemma
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
isaprop_isbracketMor
null
ishssMor {T T' : hss} (β : algebra_mor _ T T') : UU := isbracketMor β.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
ishssMor
A morphism of hss is a pointed morphism that is compatible with both [τ] and [fbracket]
hssMor (T T' : hss) : UU := ∑ β : algebra_mor _ T T', ishssMor β.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
hssMor
null
isAlgMor_hssMor {T T' : hss} (β : hssMor T T') : isAlgMor β := pr1 (pr2 β).
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
isAlgMor_hssMor
null
isbracketMor_hssMor {T T' : hss} (β : hssMor T T') : isbracketMor β := pr2 β.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
isbracketMor_hssMor
A morphism of hss is a pointed morphism that is compatible with both [τ] and [fbracket]
hssMor_eq1 : β = β' ≃ (pr1 β = pr1 β').
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
hssMor_eq1
null
hssMor_eq : β = β' ≃ (β : EndC ⟦ _ , _ ⟧) = β'.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
hssMor_eq
null
isaset_hssMor (T T' : hss) : isaset (hssMor T T').
Lemma
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
isaset_hssMor
null
ishssMor_id (T : hss) : ishssMor (identity (C:=category_FunctorAlg _) (pr1 T)).
Lemma
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
ishssMor_id
*** Identity morphism of hss
hssMor_id (T : hss) : hssMor _ _ := tpair _ _ (ishssMor_id T).
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
hssMor_id
null
ishssMor_comp {T T' T'' : hss} (β : hssMor T T') (γ : hssMor T' T'') : ishssMor (compose (C:=category_FunctorAlg _) (pr1 β) (pr1 γ)).
Lemma
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
ishssMor_comp
*** Composition of morphisms of hss
hssMor_comp {T T' T'' : hss} (β : hssMor T T') (γ : hssMor T' T'') : hssMor T T'' := tpair _ _ (ishssMor_comp β γ).
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
hssMor_comp
null
hss_obmor : precategory_ob_mor.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
hss_obmor
null
hss_precategory_data : precategory_data.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
hss_precategory_data
null
is_precategory_hss : is_precategory hss_precategory_data.
Lemma
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
is_precategory_hss
null
hss_precategory : precategory := tpair _ _ is_precategory_hss.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
hss_precategory
null
has_homsets_precategory_hss : has_homsets hss_precategory.
Lemma
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
has_homsets_precategory_hss
null
hss_category : category := hss_precategory ,, has_homsets_precategory_hss.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
hss_category
null
τ := tau_from_alg.
Notation
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
τ
null
η := eta_from_alg.
Notation
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems.v
η
null
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.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
GenMendlerIteration
Lemma 8
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'.
Theorem
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
fusion_law
Lemma 9
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} .
Lemma
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
fbracket_natural
Lemma 15
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}.
Lemma
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
compute_fbracket
null
Monad_from_hss : ∏ (C : category) (CP : BinCoproducts C) (H : Signature C C C), hss CP H → Monad C.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
Monad_from_hss
Theorem 24
hss_to_monad_functor : ∏ (C : category) (CP : BinCoproducts C) (H : Signature C C C), functor (hss_precategory CP H) (category_Monad C).
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
hss_to_monad_functor
Theorem 25
faithful_hss_to_monad : ∏ (C : category) (CP : BinCoproducts C) (H : Signature C C C), faithful (hss_to_monad_functor C CP H).
Lemma
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
faithful_hss_to_monad
Lemma 26
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) ⟧.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
bracket_for_initial_algebra
Theorem 28 in three steps: - the operation itself - its compatibility with variables - its compatibility with signature-dependent constructions
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.
Lemma
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
bracket_Thm15_ok_η
null
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.
Lemma
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
bracket_Thm15_ok_τ
null
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).
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
Initial_HSS
Theorem 29
Sum_of_Signatures : ∏ (C D D': category), BinCoproducts D → Signature C D D' → Signature C D D' → Signature C D D'.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
Sum_of_Signatures
Lemma 30
App_Sig : ∏ (C : category), BinProducts C → Signature C C C.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
App_Sig
Definition 31
Lam_Sig : ∏ (C : category), Terminal C → BinCoproducts C → BinProducts C → Signature C C C.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
Lam_Sig
Definition 32
Flat_Sig : ∏ (C : category), Signature C C C.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
Flat_Sig
Definition 33
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) ⟧.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
Lam_Flatten
Definition 36
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) ⟧.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
fbracket_for_LamE_algebra_on_Lam
Lemma 37, construction of the bracket
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 ⟧.
Definition
SubstitutionSystems
[ "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath", "UniMath" ]
UniMath/SubstitutionSystems/SimplifiedHSS/SubstitutionSystems_Summary.v
EVAL
Morphism from initial hss to construed hss, consequence of Lemma 37