38 lines
1.7 KiB
Text
38 lines
1.7 KiB
Text
import macros
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definition reflexive {A : TypeU} (R : A → A → Bool) := ∀ x, R x x
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definition transitive {A : TypeU} (R : A → A → Bool) := ∀ x y z, R x y → R y z → R x z
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definition subrelation {A : TypeU} (R1 : A → A → Bool) (R2 : A → A → Bool) := ∀ x y, R1 x y → R2 x y
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infix 50 ⊆ : subrelation
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-- (tcls R) is the transitive closure of relation R
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-- We define it as the intersection of all transitive relations containing R
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definition tcls {A : TypeU} (R : A → A → Bool) (a b : A)
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:= ∀ P, (reflexive P) → (transitive P) → (R ⊆ P) → P a b
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notation 65 _⁺ : tcls -- use superscript + as notation for transitive closure
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theorem tcls_trans {A : TypeU} {R : A → A → Bool} {a b c : A} (Hab : R⁺ a b) (Hbc : R⁺ b c) : R⁺ a c
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:= take P, assume Hrefl Htrans Hsub,
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let Pab : P a b := Hab P Hrefl Htrans Hsub,
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Pbc : P b c := Hbc P Hrefl Htrans Hsub
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in Htrans a b c Pab Pbc
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theorem tcls_refl {A : TypeU} (R : A → A → Bool) : ∀ a, R⁺ a a
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:= take a P, assume Hrefl Htrans Hsub,
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Hrefl a
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theorem tcls_sub {A : TypeU} (R : A → A → Bool) : R ⊆ R⁺
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:= take a b,
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assume Hab : R a b,
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show R⁺ a b, from
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take P, assume Hrefl Htrans Hsub,
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Hsub a b Hab
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theorem tcls_step {A : TypeU} {R : A → A → Bool} {a b c : A} (H1 : R a b) (H2 : R⁺ b c) : R⁺ a c
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:= take P, assume Hrefl Htrans Hsub,
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Htrans a b c (Hsub a b H1) (H2 P Hrefl Htrans Hsub)
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theorem tcls_smallest {A : TypeU} (R : A → A → Bool) : ∀ P, (reflexive P) → (transitive P) → (R ⊆ P) → (R⁺ ⊆ P)
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:= take P, assume Hrefl Htrans Hsub,
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take a b, assume H : R⁺ a b,
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show P a b, from H P Hrefl Htrans Hsub
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