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Work on kleene
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1 changed files with 149 additions and 2 deletions
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@ -47,6 +47,9 @@ Given a pointed object A (i.e. there exists a morphism !! : ⊤ ⇒ A), (f : X
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p-rec : ∀ {X Y : Obj} → X ⇒ Y → Y × X × N ⇒ Y → X × N ⇒ Y
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p-rec {X} {Y} f g = π₁ ∘ φ' f g
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φ'-charac : ∀ {X Y : Obj} (f : X ⇒ Y) (g : Y × X × N ⇒ Y) → φ' f g ≈ ⟨ p-rec f g , ⟨ π₁ , π₂ ⟩ ⟩
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φ'-charac = {! !}
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p-rec-IB : ∀ {X Y : Obj} (f : X ⇒ Y) (g : Y × X × N ⇒ Y) → p-rec f g ∘ ⟨ idC , z ∘ ! ⟩ ≈ f
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p-rec-IB {X} {Y} f g = (pullʳ (sym commute₁)) ○ project₁
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@ -85,9 +88,13 @@ Given a pointed object A (i.e. there exists a morphism !! : ⊤ ⇒ A), (f : X
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open strongK using (strengthen)
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_↓ : ∀ {X Y} (f : X ⇒ K.₀ Y) → X ⇒ K.₀ X
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_↓ f = K.₁ π₁ ∘ τ _ ∘ ⟨ idC , f ⟩
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↓-cong : ∀ {X Y} {f g : X ⇒ K.₀ Y} → f ≈ g → f ↓ ≈ g ↓
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↓-cong eq = refl⟩∘⟨ refl⟩∘⟨ ⟨⟩-cong₂ refl eq
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_⇂_ : ∀ {X Y Z} (f : X ⇒ K.₀ Y) (g : X ⇒ K.₀ Z) → X ⇒ K.₀ Y
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_⇂_ {X} {Y} {Z} f g = extend π₁ ∘ τ (K.₀ Y , Z) ∘ ⟨ f , g ⟩
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⇂-cong₂ : ∀ {X Y Z} {f h : X ⇒ K.₀ Y} {g i : X ⇒ K.₀ Z} → f ≈ h → g ≈ i → f ⇂ g ≈ h ⇂ i
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⇂-cong₂ eq₁ eq₂ = refl⟩∘⟨ refl⟩∘⟨ ⟨⟩-cong₂ eq₁ eq₂
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restrict-η : ∀ {X Y} (f : X ⇒ K.₀ Y) → f ↓ ≈ η.η _ ⇂ f
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restrict-η {X} {Y} f = begin
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@ -110,6 +117,35 @@ Given a pointed object A (i.e. there exists a morphism !! : ⊤ ⇒ A), (f : X
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(μ.η Y ∘ (K.₁ f ∘ K.₁ π₁)) ∘ τ (X , Z) ∘ ⟨ idC , g ⟩ ≈⟨ sym-assoc ⟩∘⟨refl ○ assoc ⟩
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extend f ∘ K.₁ π₁ ∘ τ _ ∘ ⟨ idC , g ⟩ ∎
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⇂-assoc : ∀ {X Y Z A} {f : X ⇒ K.₀ Y} {g : X ⇒ K.₀ Z} {h : X ⇒ K.₀ A} → (f ⇂ g) ⇂ h ≈ f ⇂ (g ⇂ h)
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⇂-assoc {X} {Y} {Z} {A} {f} {g} {h} = begin
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(f ⇂ g) ⇂ h ≈⟨ ⇂-cong₂ (restrict-law f g) refl ⟩
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((extend f ∘ g ↓) ⇂ h) ≈⟨ restrict-law (extend f ∘ (g ↓)) h ⟩
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extend (extend f ∘ g ↓) ∘ h ↓ ≈⟨ pushˡ kleisliK.assoc ⟩
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extend f ∘ extend (g ↓) ∘ h ↓ ≈˘⟨ refl⟩∘⟨ {! !} ⟩ -- TODO RST₃
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extend f ∘ (extend g ∘ h ↓) ↓ ≈˘⟨ refl⟩∘⟨ ↓-cong (restrict-law g h) ⟩
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extend f ∘ (g ⇂ h) ↓ ≈˘⟨ restrict-law f (g ⇂ h) ⟩
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f ⇂ (g ⇂ h) ∎
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⇂∘ : ∀ {A X Y Z} {f : X ⇒ K.₀ Y} {g : X ⇒ K.₀ Z} {h : A ⇒ X} → (f ⇂ g) ∘ h ≈ (f ∘ h) ⇂ (g ∘ h)
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⇂∘ {A} {X} {Y} {Z} {f} {g} {h} = pullʳ (pullʳ ⟨⟩∘)
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extend-⇂ : ∀ {X Y Z} {f : X ⇒ K.₀ Y} {g : X ⇒ K.₀ Z} → extend (f ⇂ g) ≈ (extend f) ⇂ (extend g)
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extend-⇂ {X} {Y} {Z} {f} {g} = begin
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extend (f ⇂ g) ≈⟨ kleisliK.extend-≈ (restrict-law f g) ⟩
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extend (extend f ∘ (g ↓)) ≈⟨ kleisliK.assoc ⟩
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extend f ∘ extend (g ↓) ≈⟨ {! !} ⟩
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{! !} ≈⟨ {! !} ⟩
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{! !} ≈⟨ {! !} ⟩
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{! !} ≈˘⟨ {! kleisliK.sym-assoc !} ⟩
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extend (extend f) ∘ ((extend g) ↓) ≈˘⟨ restrict-law (extend f) (extend g) ⟩
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(extend f ⇂ extend g) ∎
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-- extend (extend π₁ ∘ τ (K.₀ Y , Z) ∘ ⟨ f , g ⟩) ≈⟨ kleisliK.assoc ⟩
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-- extend π₁ ∘ extend (τ (K.₀ Y , Z) ∘ ⟨ f , g ⟩) ≈⟨ {! !} ⟩
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-- {! !} ≈⟨ {! !} ⟩
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-- {! !} ≈⟨ {! !} ⟩
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-- extend π₁ ∘ τ (K.₀ Y , Z) ∘ ⟨ extend f , extend g ⟩ ∎
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dom-η : ∀ {X Y} (f : X ⇒ K.₀ Y) → (η.η _ ∘ f) ↓ ≈ η.η _
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dom-η {X} {Y} f = begin
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K.₁ π₁ ∘ τ _ ∘ ⟨ idC , η.η _ ∘ f ⟩ ≈˘⟨ refl⟩∘⟨ refl⟩∘⟨ (⁂∘⟨⟩ ○ ⟨⟩-cong₂ identity² refl) ⟩
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@ -121,6 +157,37 @@ Given a pointed object A (i.e. there exists a morphism !! : ⊤ ⇒ A), (f : X
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_⊑_ : ∀ {X Y} (f g : X ⇒ K.₀ Y) → Set e
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f ⊑ g = f ≈ g ⇂ f
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⊑∘ˡ : ∀ {A X Y} {f g : X ⇒ K.₀ Y} {h : A ⇒ X} → f ⊑ g → (f ∘ h) ⊑ (g ∘ h)
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⊑∘ˡ {A} {X} {Y} {f} {g} {h} leq = begin
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f ∘ h ≈⟨ leq ⟩∘⟨refl ⟩
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(g ⇂ f) ∘ h ≈⟨ ⇂∘ ⟩
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((g ∘ h) ⇂ (f ∘ h)) ∎
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extend-⊑ : ∀ {X Y} {f g : X ⇒ K.₀ Y} → f ⊑ g → extend f ⊑ extend g
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extend-⊑ {X} {Y} {f} {g} fg = begin
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extend f ≈⟨ kleisliK.extend-≈ fg ⟩
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extend (g ⇂ f) ≈⟨ extend-⇂ ⟩
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(extend g ⇂ extend f) ∎
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⊑-cong₂ : ∀ {X Y} {f g h i : X ⇒ K.₀ Y} → f ≈ h → g ≈ i → f ⊑ g → h ⊑ i
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⊑-cong₂ eq₁ eq₂ leq = (sym eq₁) ○ leq ○ ⇂-cong₂ eq₂ eq₁
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⊑-refl : ∀ {X Y} {f : X ⇒ K.₀ Y} → f ⊑ f
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⊑-refl {X} {Y} {f} = sym (begin
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extend π₁ ∘ τ _ ∘ ⟨ f , f ⟩ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ sym Δ∘ ⟩
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extend π₁ ∘ τ _ ∘ Δ ∘ f ≈⟨ refl⟩∘⟨ (pullˡ equationalLifting) ⟩
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extend π₁ ∘ K.₁ ⟨ η.η _ , idC ⟩ ∘ f ≈⟨ pullˡ (extend∘F₁ monadK π₁ ⟨ η.η Y , idC ⟩) ⟩
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extend (π₁ ∘ ⟨ η.η _ , idC ⟩) ∘ f ≈⟨ (kleisliK.extend-≈ project₁) ⟩∘⟨refl ⟩
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extend (η.η _) ∘ f ≈⟨ elimˡ kleisliK.identityˡ ⟩
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f ∎)
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⊑-trans : ∀ {X Y} {f g h : X ⇒ K.₀ Y} → f ⊑ g → g ⊑ h → f ⊑ h
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⊑-trans {X} {Y} {f} {g} {h} leq₁ leq₂ = begin
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f ≈⟨ leq₁ ⟩
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(g ⇂ f) ≈⟨ ⇂-cong₂ leq₂ refl ⟩
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((h ⇂ g) ⇂ f) ≈⟨ ⇂-assoc ○ ⇂-cong₂ refl (sym leq₁) ⟩
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(h ⇂ f) ∎
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dom-⊑ : ∀ {X Y} (f : X ⇒ K.₀ Y) → (f ↓) ⊑ (η.η _)
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dom-⊑ {X} {Y} f = begin
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(f ↓) ≈⟨ refl ⟩
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@ -172,7 +239,7 @@ Given a pointed object A (i.e. there exists a morphism !! : ⊤ ⇒ A), (f : X
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i₂ # ∘ ! ∎
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kleene₁ : ∀ {X Y} (f : X ⇒ K.₀ Y + X) → ([ i₂ # , f ]#⟩) ⊑ (f # ∘ π₁)
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kleene₁ {X} {Y} f = p-induction ((i₂ #) ∘ !) ([ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ f ∘ π₁)) (((f #) ∘ π₁) ⇂ [ i₂ # , f ]#⟩) IB {! !}
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kleene₁ {X} {Y} f = p-induction ((i₂ #) ∘ !) ([ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ f ∘ π₁)) (((f #) ∘ π₁) ⇂ [ i₂ # , f ]#⟩) IB IS
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where
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IB : i₂ # ∘ ! ≈ (extend π₁ ∘ τ _ ∘ ⟨ ((f #) ∘ π₁) , [ i₂ # , f ]#⟩ ⟩) ∘ ⟨ idC , z ∘ ! ⟩
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IB = sym (begin
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@ -183,8 +250,88 @@ Given a pointed object A (i.e. there exists a morphism !! : ⊤ ⇒ A), (f : X
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extend π₁ ∘ (i₂ # ∘ !) ∘ ⟨ f # , idC ⟩ ≈⟨ pullˡ (∘-right-strict π₁) ⟩
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(i₂ # ∘ !) ∘ ⟨ f # , idC ⟩ ≈⟨ pullʳ (sym (!-unique (! ∘ ⟨ f # , idC ⟩))) ⟩
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(i₂ #) ∘ ! ∎)
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IS : ([ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ f ∘ π₁)) ∘ ⟨ ((f #) ∘ π₁) ⇂ [ i₂ # , f ]#⟩ , idC ⟩ ≈ (((f #) ∘ π₁) ⇂ [ i₂ # , f ]#⟩) ∘ (idC ⁂ s)
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IS = begin
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([ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ f ∘ π₁)) ∘ ⟨ ((f #) ∘ π₁) ⇂ [ i₂ # , f ]#⟩ , idC ⟩ ≈⟨ pullʳ (pullʳ ⁂∘⟨⟩) ⟩
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[ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ ⟨ idC ∘ ((f #) ∘ π₁) ⇂ [ i₂ # , f ]#⟩ , (f ∘ π₁) ∘ idC ⟩ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ ⟨⟩-cong₂ identityˡ identityʳ ⟩
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[ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ ⟨ extend π₁ ∘ τ _ ∘ ⟨ (f #) ∘ π₁ , [ i₂ # , f ]#⟩ ⟩ , f ∘ π₁ ⟩ ≈⟨ {! !} ⟩
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[ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ ⟨ extend π₁ ∘ τ _ ∘ ((f #) ∘ π₁ ⁂ idC) ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ , f ∘ π₁ ⟩ ≈⟨ {! !} ⟩
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[ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ (extend π₁ ∘ τ _ ∘ ((f #) ∘ π₁ ⁂ idC) ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ⁂ idC) ∘ ⟨ idC , f ∘ π₁ ⟩ ≈⟨ {! !} ⟩
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[ π₂ ∘ (extend π₁ ∘ τ _ ∘ ((f #) ∘ π₁ ⁂ idC) ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ⁂ idC) , π₁ ∘ (extend π₁ ∘ τ _ ∘ ((f #) ∘ π₁ ⁂ idC) ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ⁂ idC) ] ∘ distributeˡ⁻¹ ∘ ⟨ idC , f ∘ π₁ ⟩ ≈⟨ {! !} ⟩
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[ π₂ , (extend π₁ ∘ τ _ ∘ ((f #) ∘ π₁ ⁂ idC) ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩) ∘ π₁ ] ∘ distributeˡ⁻¹ ∘ ⟨ idC , f ∘ π₁ ⟩ ≈⟨ {! !} ⟩
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[ π₂ , extend π₁ ∘ τ _ ] ∘ (idC +₁ ((f #) ∘ π₁ ⁂ idC) ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁) ∘ distributeˡ⁻¹ ∘ ⟨ idC , f ∘ π₁ ⟩ ≈⟨ {! !} ⟩
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[ π₂ {A = X × N} , extend π₁ ∘ τ _ ] ∘ (idC +₁ ((f #) ∘ π₁ ⁂ idC)) ∘ (idC +₁ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁) ∘ distributeˡ⁻¹ ∘ ⟨ idC , f ∘ π₁ ⟩ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ (sym-assoc ○ (sym helper) ⟩∘⟨refl ○ assoc) ⟩
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[ π₂ {A = X × N} , extend π₁ ∘ τ _ ] ∘ (idC +₁ ((f #) ∘ π₁ ⁂ idC)) ∘ i₂ ∘ ⟨ π₁ , ([ π₂ , [ i₂ # , f ]#⟩ ∘ π₁ ] ∘ distributeˡ⁻¹) ⟩ ∘ ⟨ idC , f ∘ π₁ ⟩ ≈⟨ {! !} ⟩
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[ π₂ {A = X × N} , extend π₁ ∘ τ _ ] ∘ (idC +₁ ((f #) ∘ π₁ ⁂ idC)) ∘ i₂ ∘ ⟨ π₁ ∘ ⟨ idC , f ∘ π₁ ⟩ , ([ π₂ , [ i₂ # , f ]#⟩ ∘ π₁ ] ∘ distributeˡ⁻¹) ∘ ⟨ idC , f ∘ π₁ ⟩ ⟩ ≈⟨ {! !} ⟩
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[ π₂ {A = X × N} , extend π₁ ∘ τ _ ] ∘ (idC +₁ ((f #) ∘ π₁ ⁂ idC)) ∘ i₂ ∘ ⟨ idC , ([ π₂ , [ i₂ # , f ]#⟩ ∘ π₁ ] ∘ distributeˡ⁻¹) ∘ ⟨ idC , f ∘ π₁ ⟩ ⟩ ≈⟨ {! !} ⟩
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[ π₂ {A = X × N} , extend π₁ ∘ τ _ ] ∘ i₂ ∘ ((f #) ∘ π₁ ⁂ idC) ∘ ⟨ idC , ([ π₂ , [ i₂ # , f ]#⟩ ∘ π₁ ] ∘ distributeˡ⁻¹) ∘ ⟨ idC , f ∘ π₁ ⟩ ⟩ ≈⟨ {! !} ⟩
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[ π₂ {A = X × N} , extend π₁ ∘ τ _ ] ∘ i₂ ∘ ⟨ (f #) ∘ π₁ , ([ π₂ , [ i₂ # , f ]#⟩ ∘ π₁ ] ∘ distributeˡ⁻¹) ∘ ⟨ idC , f ∘ π₁ ⟩ ⟩ ≈⟨ pullˡ inject₂ ○ assoc ⟩
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extend π₁ ∘ τ _ ∘ ⟨ (f #) ∘ π₁ , ([ π₂ , [ i₂ # , f ]#⟩ ∘ π₁ ] ∘ distributeˡ⁻¹) ∘ ⟨ idC , f ∘ π₁ ⟩ ⟩ ≈⟨ {! !} ⟩
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extend π₁ ∘ τ _ ∘ ⟨ (f #) ∘ π₁ , [ π₂ , π₁ ] ∘ ((([ i₂ # , f ]#⟩ ⁂ idC) +₁ ([ i₂ # , f ]#⟩ ⁂ idC)) ∘ distributeˡ⁻¹) ∘ ⟨ idC , f ∘ π₁ ⟩ ⟩ ≈⟨ {! !} ⟩
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extend π₁ ∘ τ _ ∘ ⟨ (f #) ∘ π₁ , [ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ ([ i₂ # , f ]#⟩ ⁂ idC) ∘ ⟨ idC , f ∘ π₁ ⟩ ⟩ ≈⟨ {! !} ⟩
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extend π₁ ∘ τ _ ∘ ⟨ (f #) ∘ π₁ , [ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ ⟨ [ i₂ # , f ]#⟩ , f ∘ π₁ ⟩ ⟩ ≈⟨ {! !} ⟩
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extend π₁ ∘ τ _ ∘ ⟨ (f #) ∘ idC ∘ π₁ , [ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ ⟨ idC ∘ [ i₂ # , f ]#⟩ , f ∘ π₁ ∘ ⟨ π₁ , π₂ ⟩ ⟩ ⟩ ≈˘⟨ {! !} ⟩
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extend π₁ ∘ τ _ ∘ ⟨ (f #) ∘ idC ∘ π₁ , ([ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ f ∘ π₁)) ∘ φ' _ _ ⟩ ≈˘⟨ refl⟩∘⟨ refl⟩∘⟨ ⟨⟩-cong₂ (pullʳ π₁∘⁂) (p-rec-IS ((i₂ #) ∘ !) ([ π₂ , π₁ ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ f ∘ π₁))) ⟩
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(extend π₁ ∘ τ _ ∘ ⟨ ((f #) ∘ π₁) ∘ (idC ⁂ s) , [ i₂ # , f ]#⟩ ∘ (idC ⁂ s) ⟩) ≈˘⟨ pullʳ (pullʳ ⟨⟩∘) ⟩
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(extend π₁ ∘ τ _ ∘ ⟨ (f #) ∘ π₁ , [ i₂ # , f ]#⟩ ⟩) ∘ (idC ⁂ s) ≈⟨ refl ⟩
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(((f #) ∘ π₁) ⇂ [ i₂ # , f ]#⟩) ∘ (idC ⁂ s) ∎
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where
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helper : i₂ ∘ ⟨ π₁ , [ π₂ , [ i₂ # , f ]#⟩ ∘ π₁ ] ∘ distributeˡ⁻¹ ⟩ ≈ (idC +₁ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁) ∘ distributeˡ⁻¹
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helper = begin
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i₂ ∘ ⟨ π₁ , [ π₂ , [ i₂ # , f ]#⟩ ∘ π₁ ] ∘ distributeˡ⁻¹ ⟩ ≈⟨ {! !} ⟩
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i₂ ∘ ⟨ π₁ ∘ distributeˡ ∘ distributeˡ⁻¹ , [ π₂ , [ i₂ # , f ]#⟩ ∘ π₁ ] ∘ distributeˡ⁻¹ ⟩ ≈⟨ {! !} ⟩
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i₂ ∘ ⟨ π₁ ∘ distributeˡ , [ π₂ , [ i₂ # , f ]#⟩ ∘ π₁ ] ⟩ ∘ distributeˡ⁻¹ ≈⟨ refl⟩∘⟨ ((⟨⟩-cong₂ (∘[] ○ []-cong₂ (π₁∘⁂ ○ identityˡ) (π₁∘⁂ ○ identityˡ)) refl) ⟩∘⟨refl) ⟩
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i₂ ∘ ⟨ [ π₁ , π₁ ] , [ π₂ , [ i₂ # , f ]#⟩ ∘ π₁ ] ⟩ ∘ distributeˡ⁻¹ ≈⟨ {! !} ⟩
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i₂ ∘ ⟨ [ π₁ , π₁ ] , [ π₂ , π₂ ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ⟩ ∘ distributeˡ⁻¹ ≈⟨ {! !} ⟩ -- pullˡ (sym ([]-unique sub₁ {! !})) ⟩
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(distributeˡ⁻¹ ∘ (idC ⁂ i₂)) ∘ ⟨ [ π₁ , π₁ ] , [ π₂ , π₂ ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ⟩ ∘ distributeˡ⁻¹ ≈⟨ {! !} ⟩
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distributeˡ⁻¹ ∘ ⟨ [ π₁ , π₁ ] , i₂ ∘ [ π₂ , π₂ ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ⟩ ∘ distributeˡ⁻¹ ≈⟨ refl⟩∘⟨ (⟨⟩-unique subb subc ⟩∘⟨refl) ⟩
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distributeˡ⁻¹ ∘ [ idC ⁂ i₁ , ⟨ idC , i₂ ∘ [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ∘ distributeˡ⁻¹ ≈⟨ {! !} ⟩
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distributeˡ⁻¹ ∘ [ idC ⁂ i₁ , ⟨ idC ∘ idC , i₂ ∘ [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ∘ distributeˡ⁻¹ ≈⟨ {! !} ⟩
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distributeˡ⁻¹ ∘ [ (idC ⁂ i₁) ∘ idC , (idC ⁂ i₂) ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ∘ distributeˡ⁻¹ ≈˘⟨ pullʳ (pullˡ []∘+₁) ⟩
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(distributeˡ⁻¹ ∘ distributeˡ) ∘ (idC +₁ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁) ∘ distributeˡ⁻¹ ≈⟨ {! !} ⟩
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(idC +₁ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁) ∘ distributeˡ⁻¹ ∎
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where
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subb : π₁ ∘ [ idC ⁂ i₁ , ⟨ idC , i₂ ∘ [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ≈ [ π₁ , π₁ ]
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subb = ∘[] ○ []-cong₂ (π₁∘⁂ ○ identityˡ) (cancelˡ project₁)
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subc : π₂ ∘ [ idC ⁂ i₁ , ⟨ idC , i₂ ∘ [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ≈ i₂ ∘ [ π₂ , π₂ ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ]
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subc = begin
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π₂ ∘ [ idC ⁂ i₁ , ⟨ idC , i₂ ∘ [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ≈⟨ ∘[] ⟩
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[ π₂ ∘ (idC ⁂ i₁) , π₂ ∘ ⟨ idC , i₂ ∘ [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ≈⟨ []-cong₂ π₂∘⁂ (pullˡ project₂) ⟩
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[ i₁ {A = K.₀ Y} {B = K.₀ Y} ∘ π₂ {A = X × N} {B = K.₀ Y} , (i₂ ∘ [ i₂ # , f ]#⟩) ∘ π₁ ] ≈⟨ []-cong₂ {! !} refl ⟩
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[ {! _♯ !} , (i₂ ∘ [ i₂ # , f ]#⟩) ∘ π₁ ] ≈⟨ {! !} ⟩
|
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[ i₂ {A = K.₀ Y} {B = K.₀ Y} ∘ π₂ {A = X × N} {B = K.₀ Y} , i₂ ∘ [ i₂ # , f ]#⟩ ∘ π₁ ] ≈⟨ sym ([]-cong₂ refl (refl⟩∘⟨ (pullˡ project₂))) ⟩
|
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[ i₂ ∘ π₂ , i₂ ∘ π₂ ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ≈⟨ sym ∘[] ⟩
|
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i₂ ∘ [ π₂ , π₂ ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ∎
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suba : ⟨ [ π₁ , π₁ ] , i₂ ∘ [ π₂ , π₂ ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ⟩ ∘ i₁ ≈ idC ⁂ i₁
|
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suba = begin
|
||||
⟨ [ π₁ , π₁ ] , i₂ ∘ [ π₂ , π₂ ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ⟩ ∘ i₁ ≈⟨ ⟨⟩∘ ⟩
|
||||
⟨ [ π₁ , π₁ ] ∘ i₁ , (i₂ ∘ [ π₂ , π₂ ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ]) ∘ i₁ ⟩ ≈⟨ ⟨⟩-cong₂ inject₁ (pullʳ inject₁) ⟩
|
||||
⟨ π₁ , i₂ ∘ π₂ ⟩ ≈⟨ ⟨⟩-cong₂ {! !} {! !} ⟩
|
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idC ⁂ i₁ ∎
|
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sub₁ : (i₂ ∘ ⟨ [ π₁ , π₁ ] , [ π₂ , π₂ ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ⟩) ∘ i₁ ≈ i₁ ∘ idC
|
||||
sub₁ = begin
|
||||
(i₂ ∘ ⟨ [ π₁ , π₁ ] , [ π₂ , π₂ ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ⟩) ∘ i₁ ≈⟨ pullʳ ⟨⟩∘ ⟩
|
||||
i₂ ∘ ⟨ [ π₁ , π₁ ] ∘ i₁ , [ π₂ , π₂ ∘ ⟨ idC , [ i₂ # , f ]#⟩ ⟩ ∘ π₁ ] ∘ i₁ ⟩ ≈⟨ refl⟩∘⟨ ⟨⟩-cong₂ inject₁ inject₁ ⟩
|
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i₂ ∘ ⟨ π₁ , π₂ ⟩ ≈⟨ elimʳ ⁂-η ⟩
|
||||
i₂ {A = (X × N) × K.₀ Y} {B = (X × N) × K.₀ Y} ≈⟨ {! !} ⟩
|
||||
i₁ {A = (X × N) × K.₀ Y} {B = (X × N) × K.₀ Y} ∘ idC ∎
|
||||
|
||||
kleene₂ : ∀ {X Y} (f : X ⇒ K.₀ Y + X) (g : X ⇒ K.₀ Y) → ([ i₂ # , f ]#⟩) ⊑ (g ∘ π₁) → (f #) ⊑ g
|
||||
kleene₂ = {! !}
|
||||
kleene₂ {X} {Y} f g leq = ⊑-trans (⊑-trans (⊑-cong₂ refl eq₁ ⊑-refl) leq₁) (⊑-trans (⊑-cong₂ refl eq₂ ⊑-refl) leq₂)
|
||||
where
|
||||
h : X × N ⇒ (K.₀ N) + (X × N)
|
||||
h = {! !}
|
||||
eq₁ : f # ≈ extend [ i₂ # , f ]#⟩ ∘ τ _ ∘ ⟨ idC , h # ∘ ⟨ idC , z ∘ ! ⟩ ⟩
|
||||
eq₁ = {! !}
|
||||
leq₁ : (extend [ i₂ # , f ]#⟩ ∘ τ _ ∘ ⟨ idC , h # ∘ ⟨ idC , z ∘ ! ⟩ ⟩) ⊑ (extend (g ∘ π₁) ∘ τ _ ∘ ⟨ idC , h # ∘ ⟨ idC , z ∘ ! ⟩ ⟩)
|
||||
leq₁ = ⊑∘ˡ (extend-⊑ leq)
|
||||
leq₂ : (g ⇂ (h # ∘ ⟨ idC , z ∘ ! ⟩)) ⊑ g
|
||||
leq₂ = begin
|
||||
(g ⇂ ((h #) ∘ ⟨ idC , z ∘ ! ⟩)) ≈⟨ ⇂-cong₂ ⊑-refl refl ⟩
|
||||
((g ⇂ g) ⇂ ((h #) ∘ ⟨ idC , z ∘ ! ⟩)) ≈⟨ ⇂-assoc ⟩
|
||||
(g ⇂ (g ⇂ ((h #) ∘ ⟨ idC , z ∘ ! ⟩))) ∎
|
||||
eq₂ : (extend (g ∘ π₁) ∘ τ _ ∘ ⟨ idC , h # ∘ ⟨ idC , z ∘ ! ⟩ ⟩) ≈ (g ⇂ (h # ∘ ⟨ idC , z ∘ ! ⟩))
|
||||
eq₂ = {! !}
|
||||
|
||||
|
||||
```
|
||||
|
|
|
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Loading…
Reference in a new issue