Commit 630a79b2 by Max Rapp

parent 1bcea039
 ... ... @@ -32,7 +32,12 @@ theory LeastMost : base:?Logic = prop_isLeast : {A : type} A ⟶ (A ⟶ bool) ⟶ (A ⟶ A ⟶ bool) ⟶ bool ❘ = [A,x,P,R] P x ∧ ∀[y] P y ⇒ R x y ❘ # isLeast 2 3 4 ❙ least : {A : type} (A ⟶ bool) ⟶ (A ⟶ A ⟶ bool) ⟶ A ❘ = [A,P,R] ι [x] isLeast x P R ❘ # least 2 3❙ prop_isGreatest : {A : type} A ⟶ (A ⟶ bool) ⟶ (A ⟶ A ⟶ bool) ⟶ bool ❘ = [A,x,P,R] isLeast x P (converse R) ❘ # isGreatest 2 3 4 ❙ greatest : {A : type} (A ⟶ bool) ⟶ (A ⟶ A ⟶ bool) ⟶ A ❘ = [A,P,R] ι [x] isGreatest x P R ❘ # greatest 2 3❙ greatest : {A : type} (A ⟶ bool) ⟶ (A ⟶ A ⟶ bool) ⟶ A ❘ = [A,P,R] ι [x] isGreatest x P R ❘ # greatest 2 3❙ // least/greatest assume that there is a unique least element. Instead minimal/maximal return a set of minimal elements (a singleton // if there is a unique one). ❙ minimal: {A : type} (A ⟶ bool) ⟶ (A ⟶ A ⟶ bool) ⟶ set A ❘ = [A,P,R] ι [s] ∀[x] x ∈ s ⇒ isLeast x P R ❘ # minimal 2 3❙ maximal: {A : type} (A ⟶ bool) ⟶ (A ⟶ A ⟶ bool) ⟶ set A ❘ = [A,P,R] ι [s] ∀[x] x ∈ s ⇒ isGreatest x P R ❘ # maximal 2 3❙ ❚ theory DirectedSet : base:?Logic = ... ... @@ -58,6 +63,12 @@ theory DirectedSet : base:?Logic = directedSet = Mod directedSet_theory ❘ role abbreviation ❙ ❚ theory Fixpoints : ur:?LF = include ?AllSets ❙ is_fixpoint: {A:type, f:A ⟶ A, a:A} prop ❘ = [A,f,a] f a ≐ a ❙ fixpoints: {A:type} (A ⟶ A) ⟶ set A ❘ = [A,f] ι [s] ∀[x] x ∈ s ⇒ is_fixpoint A f x ❙ ❚ theory POSet : base:?Logic = include ?PrOSet ❙ ... ... @@ -70,3 +81,4 @@ theory POSet : base:?Logic = poset = Mod poset_theory ❘ role abbreviation ❙ ❚
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