...
 
Commits (3)
......@@ -272,6 +272,8 @@ theory OptionType : ur:?PLF =
None : {A : 𝒰 100} Option A ❘ # None ❙
None_is_not_Defined : {A : 𝒰 100} ⊦ isDefined (None A) ≐ false ❙
Not_None_is_Defined : {A, a : Option A} ⊦ (a ≠ None A) ⟶ ⊦ isDefined a ❙
Some : {A : 𝒰 100} A ⟶ Option A ❙
theory DescriptionOperator : ur:?PLF =
......
......@@ -33,22 +33,41 @@ theory Regularity : fnd:?Logic =
theory Specification : fnd:?Logic =
include ?preAxioms ❙
include fnd:?DescriptionOperator ❙
include fnd:?NaturalDeduction ❙
axiom_schema_specification : {P : set ⟶ prop} ⊦ ∀ [A] ∃ [M] ∀[x] (x ∈ M ⇔ x ∈ A ∧ P x ) ❙ //T also called axiom of separation ❙
axiom_schema_specification : {P : set ⟶ prop} ⊦ ∀ [A] ∃ [M] ∀[x] (x ∈ M ⇔ x ∈ A ∧ P x ) ❙ //T also called axiom of separation ❙ // here P is used as an LF parameter because unlike the ❙
theorem_specified_set_is_unique : {P : set ⟶ prop} ⊦ ∀ [A] ∃! [M] ∀[x] (x ∈ M ⇔ x ∈ A ∧ P x ) ❘# theorem_specified_set_is_unique 1 ❙ // using extensionality axiom ❙
set_builder : set ⟶ (set ⟶ prop) ⟶ set ❘ = [s][p] that set ([M] ∀[x] (x ∈ M ⇔ x ∈ s ∧ p x ) ) (forallE (theorem_specified_set_is_unique p) s) ❘# { 1 | 2 } ❙
theory Pairing : fnd:?Logic =
include ?preAxioms ❙
include ?Extensionality ❙
include fnd:?DescriptionOperator ❙
axiom_pairing : ⊦ ∀ [x] ∀ [y] ∃ [z] (x ∈ z ∧ y ∈ z)❙
theorem_pair_unique : ⊦ ∀[X] ∀ [Y] ∃! [Z] (X ∈ Z ∧ Y ∈ Z) ❙ // by axiom of extensionality❙
theorem_pair_unique_alt : {X: set} {Y :set } ⊦ ∃! [Z] (X ∈ Z ∧ Y ∈ Z) ❘ # theorem_pair_unique_alt 1 2 ❘ = [X,Y] forallE (forallE theorem_pair_unique X) Y ❙ // by axiom of extensionality❙
upair : set ⟶ set ⟶ set ❘ = [A :set][B : set] that set ([Z] (A ∈ Z ∧ B ∈ Z)) (theorem_pair_unique_alt A B) ❘ # { 1 , 2 } prec -10 ❙
singleton : set ⟶ set ❘ = [A : set] { A , A } ❘ # { 1 } prec -9❙
theory Union : fnd:?Logic =
include ?preAxioms
include ?Pairing
axiom_union : ⊦ ∀[A] ∃[B] ∀[c] (c ∈ B ⇔ ∃ [D] (c ∈ D ∧ D ∈ A )) ❙
theorem_union_unique : ⊦ ∀ [A] ∃! [B] ∀[c] (c ∈ B ⇔ ∃ [D] (c ∈ D ∧ D ∈ A )) ❙ // again using the extensionality axiom ❙
bigUnion : set ⟶ set ❘ = [S] that set ([B] ∀[c] (c ∈ B ⇔ ∃ [D] (c ∈ D ∧ D ∈ S ))) (forallE theorem_union_unique S) ❘ # ⋃ 1 ❙
binaryUnion : set ⟶ set ⟶ set ❘ = [A , B] ⋃ { A , B } ❘ # 1 ∪ 2 ❙
theory Replacement : fnd:?Logic =
......@@ -57,7 +76,7 @@ theory Replacement : fnd:?Logic =
theory Empty : fnd:?DescriptionOperator =
theory EmptySet : fnd:?DescriptionOperator =
include ?preAxioms ❙
axiom_empty_set : ⊦ ∃ [X] ∀ [y] ¬ ( y ∈ X)❙
......@@ -68,9 +87,56 @@ theory Empty : fnd:?DescriptionOperator =
theory Infinity : fnd:?Logic =
include ?Empty ❙
axiom_infinity : ⊦ ∃ [I] (∅ ∈ I ∧ ∀ [x] ∈ I ()) ❙
include ?EmptySet ❙
include ?Union ❙
axiom_infinity : ⊦ ∃ [X] ( ∅ ∈ X ∧ ∀ [y] (y ∈ X ⇒ ( y ∪ {y}) ∈ X )) ❙
prop_is_infinite : set ⟶ prop ❘ = [S] ( ∅ ∈ S ∧ ∀ [y] (y ∈ S ⇒ ( y ∪ {y}) ∈ S )) ❙
theorem_not_all_infinite_sets_are_equal : ⊦ ∃ [A] ∃ [B] prop_is_infinite A ⇒ prop_is_infinite B ⇒ A ≠ B ❙
theory PowerSet : fnd:?Logic =
include ?preAxioms ❙
include fnd:?DescriptionOperator ❙
include fnd:?NaturalDeduction ❙
is_subset : set ⟶ set ⟶ prop ❘ = [A , B] ∀[x] (x ∈ A ⇒ x ∈ B) ❘ # 1 ⊆‍ 2 ❙
axiom_power_set : ⊦ ∀ [X] ∃ [Y] ∀ [Z] (Z ⊆‍ X ⇒ Z ∈ Y) ❙
theorem_power_set_unique : ⊦ ∀ [X] ∃! [Y] ∀ [Z] (Z ⊆‍ X ⇒ Z ∈ Y) ❙
powerSet : set ⟶ set ❘= [s] that set ([Y] ∀ [Z] (Z ⊆‍ s ⇒ Z ∈ Y) ) (forallE theorem_power_set_unique s) ❙
theory ZF : fnd:?Logic =
include ?Regularity ❙
include ?Specification ❙
include ?Replacement ❙
include ?Infinity ❙
include ?PowerSet ❙
theory AxiomOfChoice : fnd:?Logic =
theory ZFC : fnd:?Logic =
include ?ZF ❙
include ?AxiomOfChoice ❙
theory TarskisAxiom : fnd:?Logic =
theory Tarski : fnd:?Logic =
theory TarskiGrothendieck : fnd:?Logic =
include ?Regularity ❙
// include ❙
\ No newline at end of file