On Models of General Type-Theoretical Languages

On Models of General Type-Theoretical Languages

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C C^∗

C

C C^∗

o o

o

o

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P T

o∈P T T Y P E_{P T} P T ⊆T Y P E_{P T}

α, β ∈T Y P E_{P T} ⇒ α, β ∈T Y P E_{P T} o

α, β α

β

α β

L=LC, V ar, Con, Cat

LC LC={λ,(,)}

V ar = ∪α∈T Y P EP TV ar(α) V ar(α)

V ar(α) α

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Con=∪α∈T Y P EP TCon(α) Con(α) Cat=∪α∈T Y P EP TCat(α) Cat(α)

V ar(α)∪Con(α)⊆Cat(α)

C∈Cat(α, β) B∈Cat(α)⇒‘C(B) ∈Cat(β) A∈Cat(β) τ ∈V ar(α)⇒‘(λτ A) ∈Cat(α, β)

F Dom_F(γ)γ∈T Y P EP T

γ∈P T Dom_F(γ)

Dom_F(α, β) =Dom_F(β)^DomF(α) α, β ∈T Y P E_{P T} P F

Dom_{P F}(γ)_{γ∈T Y P EP T}

γ ∈P T Dom_{P F}(γ)

Θγ γ Dom_{P F}(γ)\ {Θγ} =∅

Dom_{P F}(α, β) = Dom_{P F}(β)^{DomP F(α)} α, β ∈T Y P E_{P T}

Θα,β = g g ∈ Dom_{P F}(α, β) g(u) = Θβ u ∈

Dom_{P F}(α)

M G G, , v

G

, v Con V ar

a∈Con(α) (a)∈Dom_G(α) τ ∈V ar(α) v(τ)∈Dom_G(α)

M G G

Con L

Con

Cat L Cat(α) α

L(α∈T Y P EP T)

v

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M =F, , v F

Dom_M(α) =Dom_F(α). P M =P F, , v P F

Dom_{P M}(α) =Dom_{P F}(α)\ {Θα}.

M (=G, , v) ξ ∈V ar(γ) u∈Dom_G(γ) M_ξ^u(=G, , v[ξ :u]) M v[ξ :u](ξ) =u

M(=G, , v) A

α [[A]]M

a∈Con(γ) [[a]]M =(a) ξ∈V ar(γ) [[ξ]]M =v(ξ)

A∈Cat(α, β) B ∈Cat(α) [[A(B)]]M = [[A]]M([[B]]M) A β ξ ∈V ar(α) [[λξA]]M =g g

Dom_G(α) Dom_G(β) g(u) = [[A]]M_τ^u

u∈Dom_G(α)

M A∈Cat(α) [[A]]M ∈Dom_M(α) M [[A]]M ∈Dom_M(α)∪ {Θα}

M A

M A∈Cat^M_mf A∈Cat(α) α [[A]]M ∈Dom_M(α)

M A∈ Cat

A ∈ Cat M₁ = G, , v₁ M₂ = G, , v₂

L G

v₁(τ) =v₂(τ) τ ∈V(A) [[A]]M1 = [[A]]M2

A∈Cat [[A]]M v

[[A]]M = [[A]]M_τ^u τ ∈V ar(γ) u∈Dom_F(γ)

V(A) A

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M L M

A, B, C(A, B, C ∈Cat) τ (τ ∈V ar) (λτ C)(A),(λτ C)(B)∈Cat^M_mf

[[A]]M = [[B]]M ⇒[[(λτ C)(A)]]M = [[(λτ C)(B)]]M

A∈Cat B, C∈Cat(γ) M L [[B]]M = [[C]]M ⇒[[A]]M = [[A[C↓B]]]M.

M L M

B τ A M

[[B]]M =u [[A^B_τ]]M = [[A]]M_τ^u.

A∈Cat τ ∈V ar(β) B ∈Cat(β) B τ A [[(λτ A)(B)]]M = [[A^B_τ]]M

M

A ∈ Cat B, C ∈ Cat(γ) A[C↓B] (∈ Cat)

λ B C

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L(=LC, V ar, Con, Cat) M(=G, , v)

≈ Cat(⊆Cat) ≈

L Cat ≈

L ∼=L L

A∼=LB γ A, B∈Cat(γ)

M ≈M L

Cat^M_mf A≈MB ⇔[[A]]M = [[B]]M

M

≈Mc L {A : A ∈ Cat, A } ∩Cat^M_mf A≈McB ⇔[[A]]M = [[B]]M

≈ L M L ≈M

≈

L

L ∼=L

M_L

M₁ M₂ L

≈M1 ≈M2 ≈M1c ≈M2c

≈ ≈ L ≈ ≈

A, B(∈Cat) A≈B ⇔A≈B

≈ ≈ L ≈

≈ A, B(∈Cat) A≈B ⇔A≈B

M₁, M₂ L

≈M1 ≈M2

M₁, M₂ L M₁ M₂

M₁, M₂ L M₁ M₂

M (=G, , v) L τ ∈V ar(γ) u∈Dom_G M M_τ^u

M₁, M₂ L M₁, M₂

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M₁, M₂ L M₁, M₂ M₁, M₂

P F P F^t

P F P F^t

P F Dom^t_{P F}(γ)γ∈T Y P EP T

γ ∈P T Dom^t_{P F}(γ) =Dom_{P F}(γ)\ {Θγ} γ =α, β Dom^t_{P F}(γ)⊆Dom_{P F}(γ)

f ∈Dom^t_{P F}(α, β)f(u)∈Dom^t_{P F}(β) u∈Dom^t_{P F}(α) f(u) = Θβ

Dom^t_F

F Dom^t_F(γ) =Dom_F(γ) γ∈T Y P E_{P T} M(=G, , v) Dom^t_M(γ) =Dom^t_G(γ)

γ ∈T Y P E_{P T}

A γ M [[A]]M ∈

Dom^t_M(γ)

A

M L A A ∈ Con(γ) γ ∈P T A∈Cat^M_mf [[A]]M ∈Dom^t_M(γ)

A∈Cat(α, β) B ∈Cat(α) M A(B) M

≈,≈ L ≈ ≈

≈ ≈ ≈

M₁, M₂ L M₂ M₁

M₁ [[A]]M2 = [[A]]M1 A∈Cat^M_mf¹

M₁, M₂ L M₂≥M₁ ≈M2⊇≈M1

M₂≥M₁ M₂ M₁

M₁ M₂

M₁ M₂ L M₂≥M₁ M₁≥M₂

M₂ M₁ M₂≥M₁ M₂ M₁ M₁

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M L

M M M M M

M M M M

M L

L M L A, B

A, B ∈ Cat

A, B M

M A∼MB C (∈Cat)

τ (∈V ar)

(λτ C)(A)∈Cat^M_mf ⇔(λτ C)(B)∈Cat^M_mf.

M L ∼M

Cat

A∼MB ⇒A∼=LB ∼=L⊇∼M M L

A, B

M A, B∈Cat\Cat^M_mf A∼MB⇔A∼=LB

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M L

C τ (λτ C)(A) (λτ C)(B)

A B M

M L

M L

∼M ∼=L ∼M ∼=L

L

M L ∼M ∼=L

∼M ∼=L

M L M

≈M

M₁, M₂ M₁ ≈M1

M₂ A≈M1B ⇒A∼M2B A, B∈Cat

M L M

A≈MB⇒A∼MB A, B∈Cat

M (= G, , v) L M (=

G, , v) v

≈M M

M

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M

M M

M L A≈MB (A, B ∈

Cat) A∼=LB γ ∈T Y P E_{P T} A, B∈Cat(γ) M L ∼=L⊇≈M M_L≥M M₁, M₂ L M₁ M₂

M_L

M L M

∼=L⊇≈M M_L≥M

M L M

∼=L⊇≈M M_L ≥M

A, B, C ∈ Cat M L

[[B]]M = [[C]]M ⇒[[A]]M = [[A[C↓B]]]M.

A, B, C∈ Cat

[[B]]M = [[C]]M ⇒[[A]]M = [[A[C↓B]]]M,

M L

M

B, C ∈Cat B≈MC([[B]]M = [[C]]M) BMC D∈Cat, τ ∈V ar (λτ D)(B)∈Cat^M_mf (λτ D)(C) ∈/ Cat^M_mf

B, C ∈Cat(γ) γ ∈T Y P E_{P T} [[(λτ D)(B)]]M = [[(λτ D)(C)]]M α, β ∈ T Y P E_{P T} α = β B ∈Cat(α), C ∈Cat(β) A= (λξξ)(B) ξ∈V ar(α) A ∈Cat

A[C↓B]∈/Cat [[A]]M = [[A[C↓B]]]M

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M L A, B, C∈ Cat

[[B]]M = [[C]]M ⇒[[A]]M = [[A[C↓B]]]M.

M L M L

M L ∼=L⊇≈M

M L

Dom_M(γ) (γ ∈P T)

Dom_M(γ) (γ ∈ P T) Dom_M(γ) (γ ∈ T Y P E_{P T}) M (= G, , v)

Dom_M(γ) (γ ∈ T Y P E_{P T})

M u

u ∈ Dom_M(α)∩Dom_M(β) α = β τ₁ ∈ V ar(α) τ₂ ∈ V ar(β) v v(τ₁) = u = v(τ₂) M = G, , v

[[τ₁]]M = [[τ₂]]M τ₁ L τ₂ M

M

M L

Dom_M(γ) (γ ∈T Y P E_{P T}) M

A≈MB AMB A, B ∈Cat^M_mf A L B α, β ∈ T Y P E_{P T} A ∈ Cat(α) B ∈ Cat(β) α = β [[A]]M ∈ Dom_M(α) [[B]]M ∈Dom_M(β) [[A]]M = [[B]]M Dom_M(α)∩Dom_M(β)=∅

G Dom_G(γ) (γ∈P T)

M M L

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M_L ∼=L

∼=L

M

M M ≥M

M_L M

M_L ≥M

M M ≥M

M_L ≥ M M_L ≥ M A≈M B

A∼=LB γ ∈ T Y P E_{P T} A, B ∈ Cat(γ)

(λτ C)(A)∈Cat (λτ C)(B)∈Cat C∈Cat τ ∈V ar

[[(λτ C)(A)]]M = [[(λτ C)(B)]]M

(λτ C)(A)∈Cat^M_mf ⇔(λτ C)(B)∈Cat^M_mf A∼MB

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