rabe combining 10
TRANSCRIPT
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th
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M
M
M
M
M
M
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Sig
Σ ∈ Sig Sen(Σ) Mod(Σ)
|=Σ ⊆ Mod(Σ) × Sen(Σ)
ML
SigML
SenML(Σ)
p
p ∈ Σ
F ⊃ G F
M ∈ ModML(Σ) (W, ≺, V ) W
≺ ⊆ W × W V : Σ × W → {0, 1}
V Sen(Σ) × W → {0, 1}
M |=MLΣ F V (F, w) = 1 w ∈ W
(Σ, Θ)
Θ ⊆ Sen(Σ)
F ∈ Sen(Σ) (Σ, Θ) M ∈ Mod(Σ)
M |=Σ A A ∈ Θ M |=Σ F
Θ |=Σ F
Sig
Sen
Sig → SET
Mod Sig → CAT op
Sen Mod
|=Σ
σ : Σ → Σ
Σ Σ SenML(σ) :
SenML(Σ) → SenML(Σ) ModML(σ)
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(W, ≺, V
) ∈ ModML
(Σ
)
(W, ≺, V ) ∈ ModML
(Σ)
V ( p, w) = V (σ( p), w)
σ : (Σ, Θ) → (Σ, Θ) σ : Σ → Σ
A ∈ Θ Sen(σ)(A) (Σ, Θ)
Sen(σ) (Σ, Θ) (Σ, Θ)
σ Mod(σ)
(Σ, Θ) (Σ, Θ)
C
C
I
I
I
I
Θ |=Σ F
I
I
I
Th (I
)
Th (I
)
I
λ
HOL
tp : type
tm : tp → type
bool : tp
ded : tm bool → type
type tp
tp → type
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tp
tm A
A
bool
tm bool
ded
ded F
F
F
ded F
F
ded F
bool
⇒ : tm bool → tm bool → tm bool
⇒E : {F : bool } {G : bool } ded (F ⇒ G) → ded F → ded G
∀ : {A : tp} (tm A → bool ) → bool
⇒ ⇒E
F G
F ⇒ G F
G
F
G ∀
S → T
T
S
∀
A
tm A
Πx:AB(x) x : A B(x)
{x : A} B x λx:At(x) x : A t(x) [x : A] t x
A → B {x : A} B x B
K ::= type | {x : A} K
A, B ::= a | A t | [x : A] B | {x : A} B
s, t ::= c | x | [x : A] t | s t
{x : A} K {x : A} B
λ
Γ Σ K kind Γ Σ A : K
Γ Σ s : A
Σ
Γ
α β η
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Σ ::= · | Σ, c : A | Σ, a : K,
σ ::= · | σ, c := t | Σ, a := A
Γ ::= · | Γ, x : A
σ ::= · | γ, x := t
Σ
a : K
c : A
Σ
x : A
Σ Σ σ : Σ → Σ
Σ Σ σ
c : A Σ σ(c) : σ(A) a : K
σ(a) : σ(K ) σ σ Σ
σ
σ
· ΣE : F · Σ σ(E ) : σ (F ) σ
Σ
Σ
αβη
LF
LF
Σ → Σ, Σ
σ : Σ → Σ Σ → Σ, c : A
σ, c := c : Σ, c : A → Σ, c : σ(A)
c Σ
Σ Γ Γ γ : Γ → Γ
Γ Γ Σ
σ
x : A
Γ
γ (x) : γ (A)
γ
γ
γ
γ
Γ Σ E : F
Γ Σ γ (E ) : γ (F ) αβη
Σ
Σ
σ : Σ → Σ
σ(·) = · σ(Γ, x : A) = σ(Γ), x : σ(A) σ(·) = ·
σ(γ, x := t) = σ(γ ), x := σ(t)
γ : Γ → Γ Σ σ : Σ → Σ σ(γ ) : σ(Γ) → σ(Γ) σ(−)
Σ Σ
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(Sig,Sen)
Sig
Sen : Sig → SET
(Sig,Sen)
(Mod , |=) Mod : Sig → CAT op |=Σ ⊆ Sen(Σ) ×
|Mod(Σ)| σ :
Σ → Σ F ∈ Sen(Σ) M ∈ |Mod(Σ)| Mod(σ)(M ) |=Σ F M |=Σ
Sen(σ)(F )
Mod :
Sigop → CAT
(SigML,SenML)
(ModML, |=ML)
σ : Σ → Σ F ∈ SenML(Σ) (W, ≺
, V ) ∈ ModML(Σ) V (F, w) =
V (SenML(σ)(F ), w)
σ
SenML(σ)(F )
(W, ≺, V ) F (W, ≺, V )
(Pf , )
Pf
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CLASS
REL | − | :
CAT op → CLASS | − |r : CAT op → REL
SET REL PFCAT |=
|= : Sen → | − |r
◦ Mod : Sen → | − | ◦ Pf
(Sig,Sen,Mod , |=,Pf , )
CAT
Sig SE T
CAT op
PFCAT
REL
CLASS
Sen
Mod
Pf
| − |r
| − |
|=
U : SET → REL×CLASS
S (S, S ) V : C AT op × PFCAT → REL × CLASS
Sig
(U ↓ V )
|=
(Sig,Sen)
(Σ, Θ)
Σ ∈ Sig Θ ⊆ Sen(Σ) Θ (Σ, Θ)
I = (Sig,Sen,Mod , |=,Pf , )
(Σ, Θ)
Σ
F
Θ F Θ IΣ F
{F 1, . . . , F n} ⊆ Θ Pf (Σ) (IΣ F i)
n1
IΣ F
Θ
F
Θ |=IΣ F M ∈ Mod(Σ)
Θ
F
I
σ : Σ → Σ
(Σ, Θ) (Σ, Θ) F ∈ Θ Θ |=Σ Sen(σ)(F )
Θ Σ Sen(σ)(F )
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σ : (Σ, Θ) → (Σ, Θ) F ∈ Sen(Σ)
Θ |=Σ F Θ
|=Σ Sen(σ)(F )
Θ Σ F Θ
Σ Sen(σ)(F )
∅ Σ F
∅ |=Σ F Σ F Θ Σ F
Θ |=Σ F (Σ, Θ) F
I = (Sig,Sen,Mod , |=,Pf ,
) I = (Sig,Sen,Mod , |=,Pf , ) I I
(Φ, α , β , γ )
Φ : Sig → Sig α : Sen →
Sen ◦ Φ β : Mod → Mod ◦ Φ γ : Pf → Pf ◦ Φ
Σ ∈ Sig F ∈ Sen(Σ) M ∈ Mod(Φ(Σ))
β Σ(M ) |=Σ F iff M
|=Φ(Σ) αΣ(F ),
Σ ∈ Sig F ∈ Sen(Σ)
γ Σ(Σ F ) =
Φ(Σ) αΣ(F ).
LOG
β Σ CAT
op
β Σ : Mod(Φ(Σ)) → Mod(Σ) β Σ(M
)
I
I
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(Φ, α) : (Sig,Sen) → (Sig ,Sen)
(Φ, α) β : (Mod , |=) → (Mod , |=)
(Φ, α) γ : (Pf , ) → (Pf , )
Ii : Sig i → (U ↓ V )
I1 I2 Φ : Sig1 → Sig2
I1
I2 ◦ Φ
LOG
(U ↓ V )
2
2
µi = (Φi, αi, β i, γ i) :
I → I i = 1, 2 µ1 µ2
m : Φ1 → Φ2
Sen(Σ)
Sen(Φ1(Σ))
Sen(Φ2(Σ))
α1Σ
α2Σ
Sen(mΣ) Mod(Σ)
Mod (Φ1(Σ))
Mod (Φ2(Σ))
β 1Σ
β 2Σ
Mod (mΣ)
Pf (Σ)
Pf (Φ1(Σ))
Pf (Φ2(Σ))
γ 1Σ
γ 2Σ
Pf (mΣ)
CAT
m (mΣ)Σ∈Sig I
mΣ : Φ1(Σ) → Φ2(Σ) µ1
µ2 α1Σ : Sen(Σ) → Sen
(Φ1(Σ)) α2Σ :
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Sen(Σ) →
Sen
(Φ
2
(Σ))
α
1
Σ Sen
(mΣ)
mΣ
I → I I I
I
I
I
I
M = I
I
M
Sig
M
M
M
M
Sig M
Sig M
M
Sig
|= (|=Σ)Σ∈Sig Φ : Sig → Sig
|= ◦ Φ (|=Φ(Σ))Σ∈Sig ◦ Φ
M
M
Sig
Φ : Sig → SigM M ◦ Φ = (Sig,SenM ◦
Φ,ModM ◦ Φ, |=M ◦ Φ,Pf M ◦ Φ, M ◦ Φ)
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Sig SigM SE T
CAT op
PFCAT
REL
CLASS
Φ SenM
ModM
Pf M
| − |r
| − |
|=M
M
Φ M ◦ Φ → M
M ◦ Φ
M
M ◦ Φ
|=M◦Φ M◦Φ
Θ M◦ΦΣ F Θ
M
Φ(Σ) F
(U ↓ V )
M : SigM → (U ↓ V ) M◦ Φ
M
I = (Sig,Sen,Mod , |=
,Pf , ) Φ : Sig → SigM
I
M
(Φ, α , β , γ ) : I → M (id , α , β , γ )
I → M ◦ Φ
I M ◦ Φ
α
β
γ
I
M ◦ Φ
LOG I
M ◦ Φ
β Σ
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M
Φi : Sigi →
SigM
i = 1, 2
Φ : Sig1 → Sig2
m : Φ1 → Φ2 ◦ Φ
(Φ,M ◦ m) = (Φ,SenM ◦ m,ModM ◦ m,Pf M ◦ m) M ◦ Φ1 M ◦ Φ2
Sig1
Sig2
SigM SET
CAT op
PFCAT
REL
CLASS
Φ1
Φ2
Φ SenM
ModM
Pf M
| − |r
| − |
m
|=M
M
mΣ
SigM
mΣ : Φ
1(Σ) → Φ2(Φ(Σ)) Σ ∈ Sig 1
SenM(mΣ) Mod
M(mΣ) Pf M(mΣ)
(Φ,M ◦ m)
(Φ,M ◦ m)
α γ
(Pf M
◦ m)Σ
(M
◦ Φ1)Σ F
= (M
◦ Φ2)Φ(Σ) (SenM
◦ m)Σ(F ).
Pf M(mΣ) (M
Φ1(Σ) F ) = M
Φ2(Φ(Σ)) SenM(mΣ)(F )
mΣ
M◦ m
M◦ Φ1 → M◦ Φ2 ◦ Φ Sig → (U ↓ V ) (Φ,M◦ m) :
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M
◦ Φ
1
→M
◦ Φ
2
(U ↓ V )
µ : I1 → I2
µi : Ii → M µ µ1 µ2
µ1 → µ2 ◦ µ
µ = (Φ, α , β , γ ) :
I1 → I2 µi = (Φi, αi, β i, γ i) : Ii → M m µ
Φ1 → Φ2 ◦ Φ
Σ ∈ Sig1
Sen1(Σ) Sen2(Φ(Σ))
SenM(Φ1(Σ)) SenM(Φ2(Φ(Σ)))
αΣ
α1Σ
SenM(mΣ)
α2Φ(Σ)
Mod1(Σ) Mod2(Φ(Σ))
ModM(Φ1(Σ)) ModM(Φ2(Φ(Σ)))
β Σ
β 1Σ
ModM(mΣ)
β 2Φ(Σ)
Pf 1(Σ) Pf 2(Φ(Σ))
Pf M(Φ1(Σ)) Pf M(Φ2(Φ(Σ)))
γ Σ
γ 1Σ
Pf M(mΣ)
γ 2Φ(Σ)
mΣ : Φ1(Σ) → Φ2(Φ(Σ))
SenM(mΣ) αΣ
α1Σ α2Φ(Σ) ModM
(mΣ) Pf M
(mΣ)
β Σ γ Σ
M = (SigM,SenM,ModM, |=M,Pf M, M)
M
M
M
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M
M
M
M
M
M
form
ded : form → type
form
Base
form : type
ded : form → type
Base Σ
base : Base → Σ base (form)
Σ base (ded) Σ
Base
Σ
Σ
Σsyn
Σmod
Σsyn
Σmod
µ : Σsyn → Σmod
Σmod
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Σ
pf
π : Σsyn → Σ pf Σ pf
Σsyn
π
Base Σsyn
Σ pf
Σmod
base
π
µ
ded
Σ pf
π(base (ded) F )
F
Γ
Σ pf
Γ
Σmod µ(base (form))
µ(base (ded))
m : Σmod → M (M, m) F
m(µ(base (ded) F ))
M
ded Σ pf Σmod
Σsyn
M
Σ = (Σsyn, Σ pf , Σmod, base , π , µ)
base (ded)
base (ded)
form
Σsyn
M
Σ
SenM(Σ) = {F | · Σsyn F : base (form)}
ML
MLsyn
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form : type
⊃ : form → form → form
: form → form
ded : form → type
⊃
base : Base → M Lsyn
p : form P Qsyn MLsyn
p : form
q : form
T = p ⊃ p ∈ SenM(P Q)
⊃
M
σ : Σ → Σ
(σsyn
, σ pf
, σmod
)
F ∈ SenM(Σ) F σsyn(base (ded) F ) = base (ded) F
SigM
Σsyn
Σ pf
Σmod
π
µ
Σsyn
Σ pf
Σmod
π
µσsyn
σmod
σ pf
σ
σsyn ◦ base = base
Base
σsyn ◦ base =
base
F = σsyn(F )
SenM(σ)(F ) = the F such that σsyn(base (ded) F ) = base (ded) F
F
σsyn
σsyn(base (o)) = base (o)
SenM(σ)(F ) = σsyn(F )
SigM
w : P Q → P Q
wsyn = id, q := q, p := q
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id
P Q
syn
w
syn
(form) = form
wsyn(ded) = ded wsyn(ded T ) = ded q ⊃ q SenM(w)(T ) =
q ⊃ q
M
(E 1, . . . , E n) x1 :
E 1, . . . , xn : E n
Γ = x1 : E 1, . . . , xm : E m Γ = x1 : F 1, . . . , xn : F n Γ
Γ
pn1 Γ Σpf pi : F i
Γ
Γ
α
M
Σ
Pf M(Σ)
Σ pf
Σ pf
Γ Γ Γ Γ
ML pf
M Lsyn πML
ML pf
mp : {x : form} {y : form} ded x ⊃ y → ded x → ded y
nec : {x : form} ded x → ded x
K : {x : form} {y : form} ded (x ⊃ y) ⊃ (x ⊃ y)
Π
{x : form}
P Q pf
M L pf
p : form
q : form
Pf M(P Q)
Γ1 = x : ded ( p ⊃ q ), y : ded p
Γ2 = z : ded q.
Pf M(P Q)
Γ1 Γ2
γ 1 = z := mp p q
mp ( p ⊃ q ) p ⊃ q (K p q ) x
y
z x y
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Σ pf
σ pf
M
σ : Σ → Σ
Pf M(σ) : Pf M(Σ) → Pf M(Σ)
Pf M(σ)(Γ) = σ pf (Γ) for Γ ∈ |Pf M(Σ)|
Pf M(σ)(γ ) = σ pf (γ ) for γ ∈ Pf M(Σ)(Γ, Γ)
w
Γ1
Pf M(w)(Γ1) = x : ded (q ⊃ q ), y : ded q
Pf M(w)(γ 1) = z := mp q q
mp (q ⊃ q ) q ⊃ q (K q q ) x
y
Pf M(P Q)
Pf M(w)(Γ1) Pf
M(w)(Γ2)
ded
Σ pf
M
Σ
MΣ F x : π(base (ded) F )
x
MPQ ( p ⊃ q ) × M
PQ p ∼= Γ1
MPQ q ∼= Γ2
Γ1
γ 1 {( p ⊃
q ), p} MPQ q Pf M(w)(γ 1) {(q ⊃ q ),q }
MPQ
q
Σmod Σmod
Σmod
Σmod
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HOL
∀ : (tm A → tm bool ) → tm bool
A
MLmod HOL
worlds : tp
acc : tm worlds → tm worlds → tm bool
worlds
acc
µML : M Lsyn → M Lmod
form := tm worlds → tm bool ⊃ := [f : tm worlds → tm bool ] [g : tm worlds → tm bool ]
[w : tm worlds ] (f w ⇒ g w)
:= [f : tm worlds → tm bool ]
[w : tm worlds ] ∀ [w : tm worlds ] (acc w w ⇒ f w)
ded := [f : tm worlds → tm bool ] ded (∀ [w : tm worlds ] f w)
µML
µML(⊃) µML()
µML(ded)
ded
ded
P Qmod M Lmod p : µ(form)
p : tm worlds → tm bool q µPQ : P Qsyn → P Qmod
µ, p := p, q := q
P Q worlds acc p q
HOL
P Qmod
HOL
M
Σ
ModM(Σ) Σmod
(M, m)
m : Σmod → M
(M, m)
(M , m)
ϕ : M → M
ϕ ◦ m = m
(M, m)
id M
ϕ : (M, m) → (M , m) ϕ : (M , m) → (M , m) ϕ ◦ ϕ
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σ
mod
M
σ : Σ → Σ
ModM(σ) : ModM(Σ) → ModM(Σ)
ModM(σ)(M, m) = (M, m ◦ σmod)
ModM(σ)(ϕ) = ϕ
ModM
Σmod Σmod
M M ϕ
m m
σmod
(HOL,m1) ∈ Mod
M(P Q)
m1
HOL
worlds N
acc ≤ N
p q odd nonzero N
base (ded) F
Σ (M, m) Σ F
(M, m) |=MΣ F iff exists t such that · M t : m(µ(base (ded) F )).
HOL true : tm bool
false : tm bool ded true ded false
(HOL,m1)
(HOL,m1) |=M
PQ F iff exists t such that · HOL t : ded (∀[w : tm N] m1(F ) w).
m1(F ) true
HOL
HOL
(HOL,m1) |=MPQ p ⊃ q m1( p ⊃ q ) w
odd w ⇒ ∀[w : tm N] ((w ≤ w) ⇒ nonzero w)
w
Σmod (M, m) (M , m) (ϕ, r)
ϕ : M → M r : ϕ ◦ m ⇒ m
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(M, m)
(Σmod, id Σmod)
F F = HOL
m ◦ µ
F
Σmod
m
M
M = (SigM,SenM,ModM, |=M,Pf M, M)
Sig F
SenM
ModM
Pf M
SigM → CAT Pf M(Σ)
Pf M(σ)
Γ1 × . . . × Γn = Γ1, . . . , Γn Γi Γi α
|=M σ : Σ → Σ F ∈ SenM(Σ)
(M, m) ∈ ModM(Σ)
ModM(σ)(M,m) |=MΣ F
M
(m ◦ σmod)µ(base (ded) F )
M
mµσsyn(base (ded) F )
M
mµbase
(ded) SenM(σ)(F )
(M,m) |=MΣ Sen
M(σ)(F )
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M
σ : Σ → Σ
F ∈ SenM
(Σ)
Pf M(σ)(MΣ F ) = x : σ pf (π(base (ded) F )) =
x : π (σsyn(base (ded) F )) = x : π(base (ded) SenM(σ)(F )) =
MΣ SenM(σ)(F )
M
M
(Σsyn, Σ pf , Σmod, base , π , µ)
ψ : Σ pf → Σmod
Σsyn
Σ pf
Σmod
π
µ
ψ
M
M
Σ
Θ
Σ
F
Σ
Θ MΣ F Θ |=M
Σ F
(M, m) ∈ ModM(Σ) (M, m) |=MΣ A A ∈ Θ
(M, m) |=MΣ F
MΣ F MΣ F 1 × . . . ×
MΣ F n
{F 1, . . . , F n} ⊆ Θ
Σ pf
t
π
base (ded) F 1 → . . . → base (ded) F n → base (ded) F
.
M
m(ψ(t))
m
µ
base (ded) F 1 → . . . → base (ded) F n → base (ded) F
.
M
ti m(µ(base (ded) F i))
M
m(ψ(t)) t1 . . . tn m(µ(base (ded) F )) (M, m) |=M
Σ F
Σmod Σ pf
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Σ
mod
→ Σ
syn
m ◦ ψ
Σsyn
Σmod
M
Θ = {F 1, . . . , F n} M (Σ, Θ)
M Σ a1 : base (ded) F 1, . . . , an : base (ded) F n
Σsyn (Σ, Θ)syn
Σsyn → (Σ, Θ)syn (Σ, Θ) pf (Σ, Θ)mod
M
M
σ : Σ → Σ (Σ, Θ)
(Σ, Θ)
σ
σ (Σ, Θ) → (Σ, Θ)
(σ, ϑ) pf
σ
σ (Σ, Θ) → (Σ, Θ)
(σ, ϑ)mod
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Σ pf Σ pf
(Σ, Θ) pf (Σ, Θ) pf
σ pf
(σ, ϑ) pf
Σmod Σmod
(Σ, Θ)mod (Σ, Θ)mod
σmod
(σ, ϑ)mod
σ
pi : σsyn(π(base (ded) F i)) (Σ
, Θ) pf
F i ∈ Θ
ai : base (ded) F i (Σ, Θ)
syn
ai pi
(σ, ϑ) pf : (Σ, Θ) pf → (Σ, Θ) pf
Σ
(M, m)
Θ m : Σmod → M (Σ, Θ)mod
m (Σ, Θ)mod
(σ, ϑ)mod Σ Σ
σ
(σ, ϑ)mod
Σ
((Σ, Θ)mod, σmod)
(Σ, Θ)mod
M
σ
M
(M, m) M
M
F
F Σmod
Σmod
F
m : Σmod → F
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F 0 F P : F 0 → F F 0
F F P
F 0
F 0
P F
F 0
F P
F 0
F 0
Σsyn
µ : Σsyn → Σmod
ZF → ZF C
F 0 SigM
F 0
SigM
Σ
F 0 Σmod
σ
σmod
F 0
SigMF 0
F 0 SigM
F 0
P : F 0 → F
ModMP SigM
F 0→ CAT op
Σ ModM(Σ) (F , m)
m
P
F 0
ModM
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Σ
(F , m) σ
Σmod Σmod
F 0
F
σmod
m ◦ σmod mP
P : F 0 → F
MP = (SigM
F 0 ,SenM
,ModM
P , |=M
,Pf M
, M
)
SenM |=M Pf M M SigMF 0 ModMP
ModMP (σ)
MP
F 0 = F = H OL P
σ :
(Σ, Θ) → (Σ, Θ) M (σ, ϑ)mod : (Σ, Θ)mod →
(Σ, Θ)mod MP
MP σ : Σ → Σ (Σ, Θ)
(Σ, Θ) σ
σ (Σ, Θ) → (Σ, Θ) MP
m : (Σ, Θ)mod → F
m : (Σ, Θ)mod → F
Σmod Σmod
(Σ, Θ)mod (Σ, Θ)mod
F
σmod
m m
σ
ModMP (σ) Θ
Θ
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σ
mod
(σ, ϑ)
mod
: (Σ, Θ)
mod
→(Σ, Θ)mod σ MP
F 0
F
F 0 = F = Z F C
Φ : Sig → SigM
M
6
3
SigM L Lsyn
Lsyn → Σsyn
L pf
Lmod
Σ pf
Σmod
LF
SigM LF
SigM
LF
SigM
SigM
(σsyn, σ pf , σmod) : Σ → Σ σsyn σ pf σmod
LF
Σ → Σ Σ Σ
SigM
LF
LF
SigM
L → Σ L
Σ Σ
LF
Lsyn → Σsyn
Σ
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Lsyn
Lsyn
Σsyn
Σsyn
lsynσsyn
L pf
L pf
Σ pf
Σ pf
πΣ
πΣ
l pf σ pf
π
π
Lsyn
Lsyn
Σsyn
Σsyn
lsynσsyn
Lmod
Lmod
lmodσmod
Σmod
Σmod
µΣ
µΣ
µ
µ
MLP M L L
MLP
P : F 0 → F
SigMF 0 L L +LF M+LF
(L, Σsyn)
(id L, σ
syn)
MLP = MP ◦ · |L+LF.
· |L+LF : L + LF → SigM
· L + LF
M
LP
ML
M
ML = (MLsyn, M L pf , MLmod,incl,incl,µML )
incl ΦML : SigML → M L +LF
Σ = { p1, . . . , pn} ∈ |Sig
ML|
σ : Σ → Σ ΦML(Σ) = (M L, Σsyn) ΦML(σ) = (id ,σsyn)
Σsyn = M Lsyn, p1 : form, . . . , pn : form
σsyn = id MLsyn , p1 := σ( p1), . . . , pn := σ( pn)
ΦML(Σ) = (Σsyn, Σ pf , Σmod,incl,incl,µΣ) ΦML(σ) = (σsyn, σ pf , σmod)
Σ pf = M L pf , p1 : form, . . . , pn : form
Σmod = M Lmod, p1 : tm worlds → tm bool , . . . , pn : tm worlds → tm bool
µΣ = µML, p1 := p1, . . . , pn := pnσ pf = id MLpf , p1 := σ( p1), . . . , pn := σ( pn)
σmod = id MLmod , p1 := σ( p1), . . . , pn := σ( pn)
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ΦML(P Q) = (P Qsyn, P Q pf , P Qmod, incl,incl, µPQ) and
ΦML(w) = (wsyn, w pf , wmod)
ΦML
(ΦML, α , β , γ ) ML
M
MLP P αΣ β Σ
m : Σmod → H OL Σ
γ Σ
γ
β
HOL
Lsyn
MLP
p : form