refactor(library/algebra/category/morphism): restore previous (and more readable) proofs
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Leonardo de Moura
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c772d7bf84
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25df44ea43
1 changed files with 67 additions and 41 deletions
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@ -57,19 +57,24 @@ namespace morphism
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theorem left_inverse_eq_right_inverse {f : a ⟶ b} {g g' : hom b a}
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(Hl : g ∘ f = id) (Hr : f ∘ g' = id) : g = g' :=
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by rewrite [-(id_right g), -Hr, assoc, Hl, id_left]
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calc
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g = g ∘ id : (id_right g)⁻¹
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... = g ∘ f ∘ g' : by rewrite -Hr
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... = (g ∘ f) ∘ g' : by rewrite assoc
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... = id ∘ g' : by rewrite Hl
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... = g' : by rewrite id_left
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theorem retraction_eq_intro [H : is_section f] (H2 : f ∘ h = id) : retraction_of f = h
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:= left_inverse_eq_right_inverse !retraction_compose H2
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theorem retraction_eq_intro [H : is_section f] (H2 : f ∘ h = id) : retraction_of f = h :=
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left_inverse_eq_right_inverse !retraction_compose H2
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theorem section_eq_intro [H : is_retraction f] (H2 : h ∘ f = id) : section_of f = h
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:= symm (left_inverse_eq_right_inverse H2 !compose_section)
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theorem section_eq_intro [H : is_retraction f] (H2 : h ∘ f = id) : section_of f = h :=
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symm (left_inverse_eq_right_inverse H2 !compose_section)
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theorem inverse_eq_intro_right [H : is_iso f] (H2 : f ∘ h = id) : f⁻¹ = h
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:= left_inverse_eq_right_inverse !inverse_compose H2
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theorem inverse_eq_intro_right [H : is_iso f] (H2 : f ∘ h = id) : f⁻¹ = h :=
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left_inverse_eq_right_inverse !inverse_compose H2
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theorem inverse_eq_intro_left [H : is_iso f] (H2 : h ∘ f = id) : f⁻¹ = h
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:= symm (left_inverse_eq_right_inverse H2 !compose_inverse)
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theorem inverse_eq_intro_left [H : is_iso f] (H2 : h ∘ f = id) : f⁻¹ = h :=
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symm (left_inverse_eq_right_inverse H2 !compose_inverse)
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theorem section_eq_retraction (f : a ⟶ b) [Hl : is_section f] [Hr : is_retraction f] :
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retraction_of f = section_of f :=
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@ -97,14 +102,24 @@ namespace morphism
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theorem composition_is_section [instance] [Hf : is_section f] [Hg : is_section g]
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: is_section (g ∘ f) :=
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is_section.mk
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(show (retraction_of f ∘ retraction_of g) ∘ g ∘ f = id,
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by rewrite [-assoc, assoc _ g f, retraction_compose, id_left, retraction_compose])
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(calc
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(retraction_of f ∘ retraction_of g) ∘ g ∘ f
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= retraction_of f ∘ retraction_of g ∘ g ∘ f : by rewrite -assoc
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... = retraction_of f ∘ (retraction_of g ∘ g) ∘ f : by rewrite (assoc _ g f)
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... = retraction_of f ∘ id ∘ f : by rewrite retraction_compose
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... = retraction_of f ∘ f : by rewrite id_left
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... = id : by rewrite retraction_compose)
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theorem composition_is_retraction [instance] [Hf : is_retraction f] [Hg : is_retraction g]
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: is_retraction (g ∘ f) :=
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is_retraction.mk
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(show (g ∘ f) ∘ section_of f ∘ section_of g = id,
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by rewrite [-assoc, {f ∘ _}assoc, compose_section, id_left, compose_section])
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(calc
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(g ∘ f) ∘ section_of f ∘ section_of g
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= g ∘ f ∘ section_of f ∘ section_of g : by rewrite -assoc
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... = g ∘ (f ∘ section_of f) ∘ section_of g : by rewrite -assoc
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... = g ∘ id ∘ section_of g : by rewrite compose_section
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... = g ∘ section_of g : by rewrite id_left
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... = id : by rewrite compose_section)
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theorem composition_is_inverse [instance] [Hf : is_iso f] [Hg : is_iso g] : is_iso (g ∘ f) :=
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!section_retraction_imp_iso
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@ -119,7 +134,7 @@ namespace morphism
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open relation
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attribute is_iso [instance]
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theorem refl (a : ob) : a ≅ a := mk id
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theorem refl (a : ob) : a ≅ a := mk id
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theorem symm ⦃a b : ob⦄ (H : a ≅ b) : b ≅ a := mk (inverse (iso H))
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theorem trans ⦃a b c : ob⦄ (H1 : a ≅ b) (H2 : b ≅ c) : a ≅ c := mk (iso H2 ∘ iso H1)
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theorem is_equivalence_eq [instance] (T : Type) : is_equivalence isomorphic :=
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@ -143,8 +158,7 @@ namespace morphism
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theorem section_is_mono [instance] (f : a ⟶ b) [H : is_section f] : is_mono f :=
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is_mono.mk
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(λ c g h H,
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calc
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(λ c g h H, calc
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g = id ∘ g : by rewrite id_left
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... = (retraction_of f ∘ f) ∘ g : by rewrite -retraction_compose
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... = (retraction_of f ∘ f) ∘ h : by rewrite [-assoc, H, -assoc]
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@ -153,8 +167,7 @@ namespace morphism
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theorem retraction_is_epi [instance] (f : a ⟶ b) [H : is_retraction f] : is_epi f :=
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is_epi.mk
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(λ c g h H,
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calc
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(λ c g h H, calc
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g = g ∘ id : by rewrite id_right
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... = g ∘ f ∘ section_of f : by rewrite -compose_section
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... = h ∘ f ∘ section_of f : by rewrite [assoc, H, -assoc]
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@ -169,20 +182,20 @@ namespace morphism
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theorem composition_is_mono [instance] [Hf : is_mono f] [Hg : is_mono g] : is_mono (g ∘ f) :=
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is_mono.mk
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(λ d h₁ h₂ H,
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have H2 : g ∘ (f ∘ h₁) = g ∘ (f ∘ h₂),
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begin
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rewrite *assoc, exact H
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end,
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mono_elim (mono_elim H2))
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have H2 : g ∘ (f ∘ h₁) = g ∘ (f ∘ h₂),
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begin
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rewrite *assoc, exact H
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end,
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mono_elim (mono_elim H2))
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theorem composition_is_epi [instance] [Hf : is_epi f] [Hg : is_epi g] : is_epi (g ∘ f) :=
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is_epi.mk
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(λ d h₁ h₂ H,
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have H2 : (h₁ ∘ g) ∘ f = (h₂ ∘ g) ∘ f,
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begin
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rewrite -*assoc, exact H
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end,
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epi_elim (epi_elim H2))
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have H2 : (h₁ ∘ g) ∘ f = (h₂ ∘ g) ∘ f,
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begin
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rewrite -*assoc, exact H
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end,
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epi_elim (epi_elim H2))
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end morphism
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namespace morphism
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--rewrite lemmas for inverses, modified from
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@ -190,23 +203,36 @@ namespace morphism
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namespace iso
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section
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variables {ob : Type} [C : category ob] include C
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variables {a b c d : ob} (f : b ⟶ a)
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(r : c ⟶ d) (q : b ⟶ c) (p : a ⟶ b)
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(g : d ⟶ c)
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variables {a b c d : ob}
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variables (f : b ⟶ a) (r : c ⟶ d) (q : b ⟶ c) (p : a ⟶ b)
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variables (g : d ⟶ c)
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variable [Hq : is_iso q] include Hq
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theorem compose_pV : q ∘ q⁻¹ = id := !compose_inverse
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theorem compose_Vp : q⁻¹ ∘ q = id := !inverse_compose
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theorem compose_V_pp : q⁻¹ ∘ (q ∘ p) = p :=
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by rewrite [assoc, inverse_compose, id_left]
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calc
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q⁻¹ ∘ (q ∘ p) = (q⁻¹ ∘ q) ∘ p : by rewrite assoc
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... = id ∘ p : by rewrite inverse_compose
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... = p : by rewrite id_left
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theorem compose_p_Vp : q ∘ (q⁻¹ ∘ g) = g :=
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by rewrite [assoc, compose_inverse, id_left]
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calc
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q ∘ (q⁻¹ ∘ g) = (q ∘ q⁻¹) ∘ g : by rewrite assoc
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... = id ∘ g : by rewrite compose_inverse
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... = g : by rewrite id_left
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theorem compose_pp_V : (r ∘ q) ∘ q⁻¹ = r :=
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by rewrite [-assoc, compose_inverse, id_right]
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calc
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(r ∘ q) ∘ q⁻¹ = r ∘ q ∘ q⁻¹ : by rewrite assoc
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... = r ∘ id : by rewrite compose_inverse
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... = r : by rewrite id_right
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theorem compose_pV_p : (f ∘ q⁻¹) ∘ q = f :=
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by rewrite [-assoc, inverse_compose, id_right]
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calc
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(f ∘ q⁻¹) ∘ q = f ∘ q⁻¹ ∘ q : by rewrite assoc
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... = f ∘ id : by rewrite inverse_compose
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... = f : by rewrite id_right
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theorem inv_pp [H' : is_iso p] : (q ∘ p)⁻¹ = p⁻¹ ∘ q⁻¹ :=
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inverse_eq_intro_left
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@ -219,12 +245,12 @@ namespace morphism
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end
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section
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variables {ob : Type} {C : category ob} include C
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variables {d c b a : ob}
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{i : b ⟶ c} {f : b ⟶ a}
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{r : c ⟶ d} {q : b ⟶ c} {p : a ⟶ b}
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{g : d ⟶ c} {h : c ⟶ b}
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{x : b ⟶ d} {z : a ⟶ c}
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{y : d ⟶ b} {w : c ⟶ a}
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variables {d c b a : ob}
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variables {i : b ⟶ c} {f : b ⟶ a}
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{r : c ⟶ d} {q : b ⟶ c} {p : a ⟶ b}
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{g : d ⟶ c} {h : c ⟶ b}
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{x : b ⟶ d} {z : a ⟶ c}
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{y : d ⟶ b} {w : c ⟶ a}
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variable [Hq : is_iso q] include Hq
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theorem moveR_Mp (H : y = q⁻¹ ∘ g) : q ∘ y = g := H⁻¹ ▸ compose_p_Vp q g
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