Pushing for the first time
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module Graph where
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open import Graph.Base public
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module Graph.Base where
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infixl 4 _≡_
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infixl 4 _≤_
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infixl 8 _+_
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infixl 9 _⇀_
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data Graph (A : Set) : Set where
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ε : Graph A
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Vertex : A → Graph A
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_+_ : Graph A → Graph A → Graph A
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_⇀_ : Graph A → Graph A → Graph A
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-- vertex axioms
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data _∈ᵥ_ { A : Set } : A → Graph A → Set where
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x∈Vx : ∀ { x : A } → x ∈ᵥ Vertex x
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+ₗ : ∀ { x : A } { G H : Graph A } → x ∈ᵥ G → x ∈ᵥ (G + H)
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+ᵣ : ∀ { x : A } { G H : Graph A } → x ∈ᵥ H → x ∈ᵥ (G + H)
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⇀ₗ : ∀ { x : A } { G H : Graph A } → x ∈ᵥ G → x ∈ᵥ (G ⇀ H)
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⇀ᵣ : ∀ { x : A } { G H : Graph A } → x ∈ᵥ H → x ∈ᵥ (G ⇀ H)
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-- edge axioms
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data ⟦_,_⟧∈ₑ_ { A : Set } : A → A → Graph A → Set where
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+ₗ : ∀ { x y : A } { G H : Graph A } → ⟦ x , y ⟧∈ₑ G → ⟦ x , y ⟧∈ₑ (G + H)
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+ᵣ : ∀ { x y : A } { G H : Graph A } → ⟦ x , y ⟧∈ₑ H → ⟦ x , y ⟧∈ₑ (G + H)
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⇀ₗ : ∀ { x y : A } { G H : Graph A } → ⟦ x , y ⟧∈ₑ G → ⟦ x , y ⟧∈ₑ (G ⇀ H)
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⇀ᵣ : ∀ { x y : A } { G H : Graph A } → ⟦ x , y ⟧∈ₑ H → ⟦ x , y ⟧∈ₑ (G ⇀ H)
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⇀ₗᵣ : ∀ { x y : A } { G H : Graph A } → x ∈ᵥ G → y ∈ᵥ H → ⟦ x , y ⟧∈ₑ (G ⇀ H)
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-- equality axioms
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record _≡_ { A : Set } ( G H : Graph A ) : Set where
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field
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vₗ : ∀ { x : A } → x ∈ᵥ G → x ∈ᵥ H
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vᵣ : ∀ { x : A } → x ∈ᵥ H → x ∈ᵥ G
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eₗ : ∀ { x y : A } → ⟦ x , y ⟧∈ₑ G → ⟦ x , y ⟧∈ₑ H
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eᵣ : ∀ { x y : A } → ⟦ x , y ⟧∈ₑ H → ⟦ x , y ⟧∈ₑ G
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record _≤_ { A : Set} ( G H : Graph A ) : Set where
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field
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def : G + H ≡ H
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refl : ∀ { A : Set } { G : Graph A } → G ≡ G
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refl = record
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{ vₗ = λ z → z ; vᵣ = λ z → z ; eₗ = λ z → z ; eᵣ = λ z → z }
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sym : ∀ { A : Set } { G H : Graph A } → G ≡ H → H ≡ G
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sym G≡H = record
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{ vₗ = _≡_.vᵣ G≡H
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; vᵣ = _≡_.vₗ G≡H
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; eₗ = _≡_.eᵣ G≡H
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; eᵣ = _≡_.eₗ G≡H
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}
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trans : ∀ { A : Set } { G H I : Graph A } → G ≡ H → H ≡ I → G ≡ I
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trans G≡H H≡I = record
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{ vₗ = λ z → _≡_.vₗ H≡I (_≡_.vₗ G≡H z)
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; vᵣ = λ z → _≡_.vᵣ G≡H (_≡_.vᵣ H≡I z)
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; eₗ = λ z → _≡_.eₗ H≡I (_≡_.eₗ G≡H z)
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; eᵣ = λ z → _≡_.eᵣ G≡H (_≡_.eᵣ H≡I z)
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}
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≡-to-≤ : ∀ { A : Set } { G H : Graph A } → G ≡ H → G ≤ H
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≡-to-≤ G≡H = record
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{ def = record
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{ vₗ = λ { (+ₗ x) → _≡_.vₗ G≡H x ; (+ᵣ x) → x }
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; vᵣ = λ z → +ₗ (_≡_.vᵣ G≡H z)
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; eₗ = λ { (+ₗ x) → _≡_.eₗ G≡H x ; (+ᵣ x) → x }
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; eᵣ = λ z → +ₗ (_≡_.eᵣ G≡H z)
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}
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}
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≤-refl : ∀ { A : Set } { G : Graph A } → G ≤ G
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≤-refl = record
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{ def = record
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{ vₗ = λ { (+ₗ x) → x ; (+ᵣ x) → x }
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; vᵣ = +ₗ
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; eₗ = λ { (+ₗ x) → x ; (+ᵣ x) → x }
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; eᵣ = +ₗ
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}
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}
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≤-trans : ∀ { A : Set } { G H I : Graph A } → G ≤ H → H ≤ I → G ≤ I
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≤-trans G≤H H≤I = record
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{ def = record
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{ vₗ = λ { (+ₗ x) → _≡_.vₗ (_≤_.def H≤I) (+ₗ (_≡_.vₗ (_≤_.def G≤H) (+ₗ x))) ; (+ᵣ x) → x }
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; vᵣ = +ᵣ
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; eₗ = λ { (+ₗ x) → _≡_.eₗ (_≤_.def H≤I) (+ₗ (_≡_.eₗ (_≤_.def G≤H) (+ₗ x))) ; (+ᵣ x) → x }
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; eᵣ = +ᵣ
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}
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}
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open import Graph.Base
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open import Graph.Properties
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open import Graph.Reasoning
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open import Data.List
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open import Function
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module Graph.Constructors where
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edge : {A : Set} → A → A → Graph A
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edge x y = (Vertex x) ⇀ (Vertex y)
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vertices : {A : Set} → List A → Graph A
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vertices = (foldr _+_ ε) ∘ (map Vertex)
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clique : {A : Set} → List A → Graph A
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clique = (foldr _⇀_ ε) ∘ (map Vertex)
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vertices[x]=Vx : { A : Set } → ∀ { x : A } → vertices [ x ] ≡ Vertex x
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vertices[x]=Vx {_} {x} = begin
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vertices [ x ] ≡⟨⟩
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Vertex x + ε ≡⟨ +-identityʳ ⟩
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Vertex x ∎
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where open ≡-Reasoning
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edgexy=clique[x,y] : { A : Set } → ∀ { x y : A } → (edge x y) ≡ (clique (x ∷ (y ∷ [])))
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edgexy=clique[x,y] {_} {x} {y} = begin
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edge x y ≡⟨⟩
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Vertex x ⇀ Vertex y ≡⟨ sym ⇀-identˡ ⟩
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(Vertex x ⇀ Vertex y) ⇀ ε ≡⟨ sym ⇀-assoc ⟩
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Vertex x ⇀ (Vertex y ⇀ ε) ≡⟨⟩
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clique (x ∷ (y ∷ [])) ∎
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where open ≡-Reasoning
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vertices≤clique : { A : Set } → ∀ ( xs : List A ) → (vertices xs) ≤ (clique xs)
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vertices≤clique [] = begin ε ∎
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where open ≤-Reasoning
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vertices≤clique (x ∷ xs) = begin
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vertices (x ∷ xs) ≡⟨⟩
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Vertex x + vertices xs ≤⟨ +⇀-order ⟩
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Vertex x ⇀ vertices xs ≤⟨ ⇀-congʳ-≤ (vertices≤clique xs) ⟩
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Vertex x ⇀ clique xs ≡⟨⟩
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clique (x ∷ xs) ∎
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where open ≤-Reasoning
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clique-++-connect : { A : Set } → ∀ ( xs ys : List A ) → clique (xs ++ ys) ≡ ((clique xs) ⇀ (clique ys))
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clique-++-connect [] ys = begin
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clique ([] ++ ys) ≡⟨⟩
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clique ys ≡⟨ sym ⇀-identʳ ⟩
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ε ⇀ clique ys ≡⟨⟩
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(clique []) ⇀ (clique ys) ∎
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where open ≡-Reasoning
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clique-++-connect (x ∷ xs) ys = begin
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clique ((x ∷ xs) ++ ys) ≡⟨⟩
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clique (x ∷ (xs ++ ys)) ≡⟨⟩
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Vertex x ⇀ clique (xs ++ ys) ≡⟨ ⇀-congʳ (clique-++-connect xs ys) ⟩
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Vertex x ⇀ (clique xs ⇀ clique ys) ≡⟨ ⇀-assoc ⟩
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(Vertex x ⇀ clique xs) ⇀ clique ys ≡⟨⟩
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clique (x ∷ xs) ⇀ clique ys ∎
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where open ≡-Reasoning
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open import Graph.Base
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open import Graph.Reasoning
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module Graph.Properties where
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+-sym : ∀ { A : Set } { G H : Graph A } → G + H ≡ H + G
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+-sym = record
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{ vₗ = λ { (+ₗ x) → +ᵣ x ; (+ᵣ x) → +ₗ x }
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; vᵣ = λ { (+ₗ x) → +ᵣ x ; (+ᵣ x) → +ₗ x }
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; eₗ = λ { (+ₗ x) → +ᵣ x ; (+ᵣ x) → +ₗ x }
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; eᵣ = λ { (+ₗ x) → +ᵣ x ; (+ᵣ x) → +ₗ x }
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}
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+-assoc : ∀ { A : Set } { G H I : Graph A } → G + (H + I) ≡ (G + H) + I
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+-assoc = record
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{ vₗ = λ { (+ₗ x) → +ₗ (+ₗ x) ; (+ᵣ (+ₗ x)) → +ₗ (+ᵣ x) ; (+ᵣ (+ᵣ x)) → +ᵣ x }
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; vᵣ = λ { (+ₗ (+ₗ x)) → +ₗ x ; (+ₗ (+ᵣ x)) → +ᵣ (+ₗ x) ; (+ᵣ x) → +ᵣ (+ᵣ x) }
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; eₗ = λ { (+ₗ x) → +ₗ (+ₗ x) ; (+ᵣ (+ₗ x)) → +ₗ (+ᵣ x) ; (+ᵣ (+ᵣ x)) → +ᵣ x }
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; eᵣ = λ { (+ₗ (+ₗ x)) → +ₗ x ; (+ₗ (+ᵣ x)) → +ᵣ (+ₗ x) ; (+ᵣ x) → +ᵣ (+ᵣ x) }
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}
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+-identityʳ : ∀ { A : Set } { G : Graph A } → G + ε ≡ G
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+-identityʳ = record
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{ vₗ = λ { (+ₗ x) → x }
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; vᵣ = λ x → +ₗ x
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; eₗ = λ { (+ₗ x) → x }
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; eᵣ = +ₗ
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}
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+-identityˡ : ∀ { A : Set } { G : Graph A } → ε + G ≡ G
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+-identityˡ = record
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{ vₗ = λ { (+ᵣ x) → x }
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; vᵣ = +ᵣ
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; eₗ = λ { (+ᵣ x) → x }
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; eᵣ = +ᵣ
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}
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+-congˡ : ∀ { A : Set } { G H I : Graph A } → G ≡ H → G + I ≡ H + I
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+-congˡ G≈H = record
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{ vₗ = λ { (+ₗ x) → +ₗ (_≡_.vₗ G≈H x) ; (+ᵣ x) → +ᵣ x }
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; vᵣ = λ { (+ₗ x) → +ₗ (_≡_.vᵣ G≈H x) ; (+ᵣ x) → +ᵣ x }
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; eₗ = λ { (+ₗ x) → +ₗ (_≡_.eₗ G≈H x) ; (+ᵣ x) → +ᵣ x }
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; eᵣ = λ { (+ₗ x) → +ₗ (_≡_.eᵣ G≈H x) ; (+ᵣ x) → +ᵣ x }
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}
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+-congʳ : ∀ { A : Set } { G H I : Graph A } → H ≡ I → G + H ≡ G + I
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+-congʳ {_} {G} {H} {I} H≡I = begin
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G + H ≡⟨ +-sym ⟩
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H + G ≡⟨ +-congˡ H≡I ⟩
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I + G ≡⟨ +-sym ⟩
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G + I ∎
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where open ≡-Reasoning
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⇀-assoc : ∀ { A : Set } { G H I : Graph A } → G ⇀ (H ⇀ I) ≡ (G ⇀ H) ⇀ I
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⇀-assoc = record
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{ vₗ = λ { (⇀ₗ x) → ⇀ₗ (⇀ₗ x) ; (⇀ᵣ (⇀ₗ x)) → ⇀ₗ (⇀ᵣ x) ; (⇀ᵣ (⇀ᵣ x)) → ⇀ᵣ x }
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; vᵣ = λ { (⇀ₗ (⇀ₗ x)) → ⇀ₗ x ; (⇀ₗ (⇀ᵣ x)) → ⇀ᵣ (⇀ₗ x) ; (⇀ᵣ x) → ⇀ᵣ (⇀ᵣ x) }
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; eₗ = λ { (⇀ₗ x) → ⇀ₗ (⇀ₗ x) ; (⇀ᵣ (⇀ₗ x)) → ⇀ₗ (⇀ᵣ x) ; (⇀ᵣ (⇀ᵣ x)) → ⇀ᵣ x ; (⇀ᵣ (⇀ₗᵣ x x₁)) → ⇀ₗᵣ (⇀ᵣ x) x₁ ; (⇀ₗᵣ x (⇀ₗ x₁)) → ⇀ₗ (⇀ₗᵣ x x₁) ; (⇀ₗᵣ x (⇀ᵣ x₁)) → ⇀ₗᵣ (⇀ₗ x) x₁ }
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; eᵣ = λ { (⇀ₗ (⇀ₗ x)) → ⇀ₗ x ; (⇀ₗ (⇀ᵣ x)) → ⇀ᵣ (⇀ₗ x) ; (⇀ₗ (⇀ₗᵣ x x₁)) → ⇀ₗᵣ x (⇀ₗ x₁) ; (⇀ᵣ x) → ⇀ᵣ (⇀ᵣ x) ; (⇀ₗᵣ (⇀ₗ x) x₁) → ⇀ₗᵣ x (⇀ᵣ x₁) ; (⇀ₗᵣ (⇀ᵣ x) x₁) → ⇀ᵣ (⇀ₗᵣ x x₁) }
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}
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⇀-identˡ : ∀ { A : Set } { G : Graph A } → G ⇀ ε ≡ G
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⇀-identˡ = record
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{ vₗ = λ { (⇀ₗ x) → x }
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; vᵣ = ⇀ₗ
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; eₗ = λ { (⇀ₗ x) → x }
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; eᵣ = ⇀ₗ
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}
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⇀-identʳ : ∀ { A : Set } { G : Graph A } → ε ⇀ G ≡ G
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⇀-identʳ = record
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{ vₗ = λ { (⇀ᵣ x) → x }
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; vᵣ = ⇀ᵣ
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; eₗ = λ { (⇀ᵣ x) → x }
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; eᵣ = ⇀ᵣ
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}
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⇀-congˡ : ∀ { A : Set } { G H I : Graph A } → G ≡ H → G ⇀ I ≡ H ⇀ I
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⇀-congˡ G≈H = record
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{ vₗ = λ { (⇀ₗ x) → ⇀ₗ (_≡_.vₗ G≈H x) ; (⇀ᵣ x) → ⇀ᵣ x }
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; vᵣ = λ { (⇀ₗ x) → ⇀ₗ (_≡_.vᵣ G≈H x) ; (⇀ᵣ x) → ⇀ᵣ x }
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; eₗ = λ { (⇀ₗ x) → ⇀ₗ (_≡_.eₗ G≈H x) ; (⇀ᵣ x) → ⇀ᵣ x ; (⇀ₗᵣ x x₁) → ⇀ₗᵣ (_≡_.vₗ G≈H x) x₁ }
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; eᵣ = λ { (⇀ₗ x) → ⇀ₗ (_≡_.eᵣ G≈H x) ; (⇀ᵣ x) → ⇀ᵣ x ; (⇀ₗᵣ x x₁) → ⇀ₗᵣ (_≡_.vᵣ G≈H x) x₁ }
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}
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⇀-congʳ : ∀ { A : Set } { G H I : Graph A } → H ≡ I → G ⇀ H ≡ G ⇀ I
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⇀-congʳ G≈H = record
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{ vₗ = λ { (⇀ₗ x) → ⇀ₗ x ; (⇀ᵣ x) → ⇀ᵣ (_≡_.vₗ G≈H x) }
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; vᵣ = λ { (⇀ₗ x) → ⇀ₗ x ; (⇀ᵣ x) → ⇀ᵣ (_≡_.vᵣ G≈H x) }
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; eₗ = λ { (⇀ₗ x) → ⇀ₗ x ; (⇀ᵣ x) → ⇀ᵣ (_≡_.eₗ G≈H x) ; (⇀ₗᵣ x x₁) → ⇀ₗᵣ x (_≡_.vₗ G≈H x₁) }
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; eᵣ = λ { (⇀ₗ x) → ⇀ₗ x ; (⇀ᵣ x) → ⇀ᵣ (_≡_.eᵣ G≈H x) ; (⇀ₗᵣ x x₁) → ⇀ₗᵣ x (_≡_.vᵣ G≈H x₁) }
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}
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⇀-congʳ-≤ : ∀ { A : Set } { G H I : Graph A } → H ≤ I → G ⇀ H ≤ G ⇀ I
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⇀-congʳ-≤ H≤I = record
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{ def = record
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{ vₗ = λ { (+ₗ (⇀ₗ x)) → ⇀ₗ x ; (+ₗ (⇀ᵣ x)) → ⇀ᵣ (_≡_.vₗ (_≤_.def H≤I) (+ₗ x)) ; (+ᵣ x) → x }
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; vᵣ = +ᵣ
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; eₗ = λ { (+ₗ (⇀ₗ x)) → ⇀ₗ x ; (+ₗ (⇀ᵣ x)) → ⇀ᵣ (_≡_.eₗ (_≤_.def H≤I) (+ₗ x)) ; (+ₗ (⇀ₗᵣ x x₁)) → ⇀ₗᵣ x (_≡_.vₗ (_≤_.def H≤I) (+ₗ x₁)) ; (+ᵣ x) → x }
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; eᵣ = +ᵣ
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}
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}
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⇀-distrib-+ : ∀ { A : Set } { G H I : Graph A } → G ⇀ (H + I) ≡ G ⇀ H + G ⇀ I
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⇀-distrib-+ = record
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{ vₗ = λ { (⇀ₗ x) → +ₗ (⇀ₗ x) ; (⇀ᵣ (+ₗ x)) → +ₗ (⇀ᵣ x) ; (⇀ᵣ (+ᵣ x)) → +ᵣ (⇀ᵣ x) }
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; vᵣ = λ { (+ₗ (⇀ₗ x)) → ⇀ₗ x ; (+ₗ (⇀ᵣ x)) → ⇀ᵣ (+ₗ x) ; (+ᵣ (⇀ₗ x)) → ⇀ₗ x ; (+ᵣ (⇀ᵣ x)) → ⇀ᵣ (+ᵣ x) }
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; eₗ = λ { (⇀ₗ x) → +ₗ (⇀ₗ x) ; (⇀ᵣ (+ₗ x)) → +ₗ (⇀ᵣ x) ; (⇀ᵣ (+ᵣ x)) → +ᵣ (⇀ᵣ x) ; (⇀ₗᵣ x (+ₗ x₁)) → +ₗ (⇀ₗᵣ x x₁) ; (⇀ₗᵣ x (+ᵣ x₁)) → +ᵣ (⇀ₗᵣ x x₁) }
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; eᵣ = λ { (+ₗ (⇀ₗ x)) → ⇀ₗ x ; (+ₗ (⇀ᵣ x)) → ⇀ᵣ (+ₗ x) ; (+ₗ (⇀ₗᵣ x x₁)) → ⇀ₗᵣ x (+ₗ x₁) ; (+ᵣ (⇀ₗ x)) → ⇀ₗ x ; (+ᵣ (⇀ᵣ x)) → ⇀ᵣ (+ᵣ x) ; (+ᵣ (⇀ₗᵣ x x₁)) → ⇀ₗᵣ x (+ᵣ x₁) }
|
||||
}
|
||||
|
||||
+-distrib-⇀ : ∀ { A : Set } { G H I : Graph A } → (G + H) ⇀ I ≡ G ⇀ I + H ⇀ I
|
||||
+-distrib-⇀ = record
|
||||
{ vₗ = λ { (⇀ₗ (+ₗ x)) → +ₗ (⇀ₗ x) ; (⇀ₗ (+ᵣ x)) → +ᵣ (⇀ₗ x) ; (⇀ᵣ x) → +ₗ (⇀ᵣ x) }
|
||||
; vᵣ = λ { (+ₗ (⇀ₗ x)) → ⇀ₗ (+ₗ x) ; (+ₗ (⇀ᵣ x)) → ⇀ᵣ x ; (+ᵣ (⇀ₗ x)) → ⇀ₗ (+ᵣ x) ; (+ᵣ (⇀ᵣ x)) → ⇀ᵣ x }
|
||||
; eₗ = λ { (⇀ₗ (+ₗ x)) → +ₗ (⇀ₗ x) ; (⇀ₗ (+ᵣ x)) → +ᵣ (⇀ₗ x) ; (⇀ᵣ x) → +ₗ (⇀ᵣ x) ; (⇀ₗᵣ (+ₗ x) x₁) → +ₗ (⇀ₗᵣ x x₁) ; (⇀ₗᵣ (+ᵣ x) x₁) → +ᵣ (⇀ₗᵣ x x₁) }
|
||||
; eᵣ = λ { (+ₗ (⇀ₗ x)) → ⇀ₗ (+ₗ x) ; (+ₗ (⇀ᵣ x)) → ⇀ᵣ x ; (+ₗ (⇀ₗᵣ x x₁)) → ⇀ₗᵣ (+ₗ x) x₁ ; (+ᵣ (⇀ₗ x)) → ⇀ₗ (+ᵣ x) ; (+ᵣ (⇀ᵣ x)) → ⇀ᵣ x ; (+ᵣ (⇀ₗᵣ x x₁)) → ⇀ₗᵣ (+ᵣ x) x₁ }
|
||||
}
|
||||
|
||||
⇀-decomp : ∀ { A : Set } { G H I : Graph A } → G ⇀ (H ⇀ I) ≡ G ⇀ H + (G ⇀ I + H ⇀ I)
|
||||
⇀-decomp = record
|
||||
{ vₗ = λ { (⇀ₗ x) → +ₗ (⇀ₗ x) ; (⇀ᵣ x) → +ᵣ (+ᵣ x) }
|
||||
; vᵣ = λ { (+ₗ (⇀ₗ x)) → ⇀ₗ x ; (+ₗ (⇀ᵣ x)) → ⇀ᵣ (⇀ₗ x) ; (+ᵣ (+ₗ (⇀ₗ x))) → ⇀ₗ x ; (+ᵣ (+ₗ (⇀ᵣ x))) → ⇀ᵣ (⇀ᵣ x) ; (+ᵣ (+ᵣ x)) → ⇀ᵣ x }
|
||||
; eₗ = λ { (⇀ₗ x) → +ₗ (⇀ₗ x) ; (⇀ᵣ x) → +ᵣ (+ᵣ x) ; (⇀ₗᵣ x (⇀ₗ x₁)) → +ₗ (⇀ₗᵣ x x₁) ; (⇀ₗᵣ x (⇀ᵣ x₁)) → +ᵣ (+ₗ (⇀ₗᵣ x x₁)) }
|
||||
; eᵣ = λ { (+ₗ (⇀ₗ x)) → ⇀ₗ x ; (+ₗ (⇀ᵣ x)) → ⇀ᵣ (⇀ₗ x) ; (+ₗ (⇀ₗᵣ x x₁)) → ⇀ₗᵣ x (⇀ₗ x₁) ; (+ᵣ (+ₗ (⇀ₗ x))) → ⇀ₗ x ; (+ᵣ (+ₗ (⇀ᵣ x))) → ⇀ᵣ (⇀ᵣ x) ; (+ᵣ (+ₗ (⇀ₗᵣ x x₁))) → ⇀ₗᵣ x (⇀ᵣ x₁) ; (+ᵣ (+ᵣ x)) → ⇀ᵣ x }
|
||||
}
|
||||
|
||||
+-idem : ∀ { A : Set } { G : Graph A } → G + G ≡ G
|
||||
+-idem {_} {G} = begin
|
||||
G + G ≡⟨ sym ⇀-identˡ ⟩
|
||||
(G + G) ⇀ ε ≡⟨ +-distrib-⇀ ⟩
|
||||
(G ⇀ ε) + (G ⇀ ε) ≡⟨ sym +-identityʳ ⟩
|
||||
((G ⇀ ε) + (G ⇀ ε)) + ε ≡⟨ sym (+-congʳ ⇀-identˡ) ⟩
|
||||
((G ⇀ ε) + (G ⇀ ε)) + (ε ⇀ ε) ≡⟨ sym +-assoc ⟩
|
||||
(G ⇀ ε) + ((G ⇀ ε) + (ε ⇀ ε)) ≡⟨ sym ⇀-decomp ⟩
|
||||
G ⇀ (ε ⇀ ε) ≡⟨ ⇀-assoc ⟩
|
||||
(G ⇀ ε) ⇀ ε ≡⟨ ⇀-identˡ ⟩
|
||||
G ⇀ ε ≡⟨ ⇀-identˡ ⟩
|
||||
G ∎
|
||||
where open ≡-Reasoning
|
||||
|
||||
+-absorp : ∀ { A : Set } { G H : Graph A } → G ⇀ H + G + H ≡ G ⇀ H
|
||||
+-absorp {_} {G} {H} = begin
|
||||
G ⇀ H + G + H ≡⟨ +-congˡ (+-congʳ (sym ⇀-identˡ)) ⟩
|
||||
G ⇀ H + G ⇀ ε + H ≡⟨ +-congʳ (sym ⇀-identˡ) ⟩
|
||||
G ⇀ H + G ⇀ ε + H ⇀ ε ≡⟨ sym +-assoc ⟩
|
||||
G ⇀ H + (G ⇀ ε + H ⇀ ε) ≡⟨ sym ⇀-decomp ⟩
|
||||
G ⇀ (H ⇀ ε) ≡⟨ ⇀-congʳ ⇀-identˡ ⟩
|
||||
G ⇀ H ∎
|
||||
where open ≡-Reasoning
|
||||
|
||||
⇀-sat : ∀ { A : Set } { G : Graph A } → G ⇀ G ⇀ G ≡ G ⇀ G
|
||||
⇀-sat {_} {G} = begin
|
||||
G ⇀ G ⇀ G ≡⟨ sym ⇀-assoc ⟩
|
||||
G ⇀ (G ⇀ G) ≡⟨ ⇀-decomp ⟩
|
||||
G ⇀ G + (G ⇀ G + G ⇀ G) ≡⟨ +-congʳ +-idem ⟩
|
||||
G ⇀ G + G ⇀ G ≡⟨ +-idem ⟩
|
||||
G ⇀ G ∎
|
||||
where open ≡-Reasoning
|
||||
|
||||
≡→≤ : ∀ { A : Set } { G H : Graph A } → G ≡ H → G ≤ H
|
||||
≡→≤ {_} {G} {H} G≡H = begin
|
||||
G ≡⟨ G≡H ⟩
|
||||
H ∎
|
||||
where open ≤-Reasoning
|
||||
|
||||
≲-least : ∀ { A : Set } { G : Graph A } → ε ≤ G
|
||||
≲-least {_} {G} = record { def = begin
|
||||
ε + G ≡⟨ +-identityˡ ⟩
|
||||
G ∎}
|
||||
where open ≡-Reasoning
|
||||
|
||||
+-order : ∀ { A : Set } { G H : Graph A } → G ≤ G + H
|
||||
+-order {_} {G} {H} = record { def = begin
|
||||
G + (G + H) ≡⟨ +-assoc ⟩
|
||||
G + G + H ≡⟨ +-congˡ +-idem ⟩
|
||||
G + H ∎}
|
||||
where open ≡-Reasoning
|
||||
|
||||
+⇀-order : ∀ { A : Set } { G H : Graph A } → G + H ≤ G ⇀ H
|
||||
+⇀-order {_} {G} {H} = record { def = begin
|
||||
G + H + G ⇀ H ≡⟨ +-congˡ +-sym ⟩
|
||||
H + G + G ⇀ H ≡⟨ +-sym ⟩
|
||||
G ⇀ H + (H + G) ≡⟨ +-congʳ +-sym ⟩
|
||||
G ⇀ H + (G + H) ≡⟨ +-assoc ⟩
|
||||
G ⇀ H + G + H ≡⟨ +-absorp ⟩
|
||||
G ⇀ H ∎}
|
||||
where open ≡-Reasoning
|
||||
|
||||
+-mono : ∀ { A : Set } { G H I : Graph A } → G ≤ H → G + I ≤ H + I
|
||||
+-mono {_} {G} {H} {I} G≤H = record { def = begin
|
||||
G + I + (H + I) ≡⟨ +-congˡ +-sym ⟩
|
||||
I + G + (H + I) ≡⟨ +-assoc ⟩
|
||||
I + G + H + I ≡⟨ +-congˡ (sym +-assoc) ⟩
|
||||
I + (G + H) + I ≡⟨ +-congˡ (+-congʳ (_≤_.def G≤H)) ⟩
|
||||
I + H + I ≡⟨ +-congˡ +-sym ⟩
|
||||
H + I + I ≡⟨ sym +-assoc ⟩
|
||||
H + (I + I) ≡⟨ +-congʳ +-idem ⟩
|
||||
H + I ∎}
|
||||
where open ≡-Reasoning
|
||||
|
||||
⇀-monoˡ : ∀ { A : Set } { G H I : Graph A } → G ≤ H → G ⇀ I ≤ H ⇀ I
|
||||
⇀-monoˡ {_} {G} {H} {I} G≤H = record { def = begin
|
||||
G ⇀ I + H ⇀ I ≡⟨ sym +-distrib-⇀ ⟩
|
||||
(G + H) ⇀ I ≡⟨ ⇀-congˡ (_≤_.def G≤H) ⟩
|
||||
H ⇀ I ∎}
|
||||
where open ≡-Reasoning
|
||||
|
||||
⇀-monoʳ : ∀ { A : Set } { G H I : Graph A } → G ≤ H → I ⇀ G ≤ I ⇀ H
|
||||
⇀-monoʳ {_} {G} {H} {I} G≤H = record { def = begin
|
||||
I ⇀ G + I ⇀ H ≡⟨ sym ⇀-distrib-+ ⟩
|
||||
I ⇀ (G + H) ≡⟨ ⇀-congʳ (_≤_.def G≤H) ⟩
|
||||
I ⇀ H ∎}
|
||||
where open ≡-Reasoning
|
||||
|
|
@ -0,0 +1,47 @@
|
|||
open import Graph.Base
|
||||
|
||||
module Graph.Reasoning where
|
||||
|
||||
|
||||
module ≡-Reasoning where
|
||||
|
||||
infix 1 begin_
|
||||
infixr 2 _≡⟨⟩_ _≡⟨_⟩_
|
||||
infix 3 _∎
|
||||
|
||||
begin_ : ∀ { A : Set } { G H : Graph A } → G ≡ H → G ≡ H
|
||||
begin G≡H = G≡H
|
||||
|
||||
_≡⟨⟩_ : ∀ { A : Set } (G : Graph A) { H : Graph A } → G ≡ H → G ≡ H
|
||||
G ≡⟨⟩ G≡H = G≡H
|
||||
|
||||
_≡⟨_⟩_ : ∀ { A : Set } (G : Graph A) { H I : Graph A } → G ≡ H → H ≡ I → G ≡ I
|
||||
G ≡⟨ G≡H ⟩ H≡I = trans G≡H H≡I
|
||||
|
||||
_∎ : ∀ { A : Set } (G : Graph A) → G ≡ G
|
||||
G ∎ = refl
|
||||
|
||||
|
||||
module ≤-Reasoning where
|
||||
|
||||
infix 1 begin_
|
||||
infixr 2 _≡⟨⟩_ _≡⟨_⟩_ _≤⟨⟩_ _≤⟨_⟩_
|
||||
infix 3 _∎
|
||||
|
||||
begin_ : ∀ { A : Set } { G H : Graph A } → G ≤ H → G ≤ H
|
||||
begin G≤H = G≤H
|
||||
|
||||
_≡⟨⟩_ : ∀ { A : Set } (G : Graph A) { H : Graph A } → G ≤ H → G ≤ H
|
||||
G ≡⟨⟩ G≤H = G≤H
|
||||
|
||||
_≡⟨_⟩_ : ∀ { A : Set } (G : Graph A) { H I : Graph A } → G ≡ H → H ≤ I → G ≤ I
|
||||
G ≡⟨ G≡H ⟩ H≤I = ≤-trans (≡-to-≤ G≡H) H≤I
|
||||
|
||||
_≤⟨⟩_ : ∀ { A : Set } (G : Graph A) { H : Graph A } → G ≤ H → G ≤ H
|
||||
G ≤⟨⟩ G≤H = G≤H
|
||||
|
||||
_≤⟨_⟩_ : ∀ { A : Set } (G : Graph A) { H I : Graph A } → G ≤ H → H ≤ I → G ≤ I
|
||||
G ≤⟨ G≤H ⟩ H≤I = ≤-trans G≤H H≤I
|
||||
|
||||
_∎ : ∀ { A : Set } (G : Graph A) → G ≤ G
|
||||
G ∎ = ≤-refl
|
||||
Loading…
Reference in New Issue