Nuprl Lemma : Formco-ext
∀[C:Type]
Formco(C) ≡ lbl:Atom × if lbl =a "Var" then Atom
if lbl =a "Const" then C
if lbl =a "Set" then var:Atom × Formco(C)
if lbl =a "Equal" then left:Formco(C) × Formco(C)
if lbl =a "Member" then element:Formco(C) × Formco(C)
if lbl =a "And" then left:Formco(C) × Formco(C)
if lbl =a "Or" then left:Formco(C) × Formco(C)
if lbl =a "Not" then Formco(C)
if lbl =a "All" then var:Atom × Formco(C)
if lbl =a "Exists" then var:Atom × Formco(C)
else Void
fi
Proof
Definitions occuring in Statement :
Formco: Formco(C),
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
product: x:A × B[x],
token: "$token",
atom: Atom,
void: Void,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
Formco: Formco(C),
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
so_apply: x[s],
continuous-monotone: ContinuousMonotone(T.F[T]),
type-monotone: Monotone(T.F[T]),
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
strong-type-continuous: Continuous+(T.F[T]),
type-continuous: Continuous(T.F[T]),
ext-eq: A ≡ B
Lemmas referenced :
corec-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
subtype_rel_product,
subtype_rel_ifthenelse,
ifthenelse_wf,
subtype_rel_wf,
strong-continuous-depproduct,
continuous-constant,
strong-continuous-product,
continuous-id,
subtype_rel_weakening,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
productEquality,
atomEquality,
hypothesisEquality,
tokenEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
voidEquality,
universeEquality,
independent_pairFormation,
axiomEquality,
isect_memberEquality,
isectEquality,
applyEquality,
functionExtensionality,
functionEquality,
independent_pairEquality
Latex:
\mforall{}[C:Type]
Formco(C) \mequiv{} lbl:Atom \mtimes{} if lbl =a "Var" then Atom
if lbl =a "Const" then C
if lbl =a "Set" then var:Atom \mtimes{} Formco(C)
if lbl =a "Equal" then left:Formco(C) \mtimes{} Formco(C)
if lbl =a "Member" then element:Formco(C) \mtimes{} Formco(C)
if lbl =a "And" then left:Formco(C) \mtimes{} Formco(C)
if lbl =a "Or" then left:Formco(C) \mtimes{} Formco(C)
if lbl =a "Not" then Formco(C)
if lbl =a "All" then var:Atom \mtimes{} Formco(C)
if lbl =a "Exists" then var:Atom \mtimes{} Formco(C)
else Void
fi
Date html generated:
2018_05_21-PM-10_41_59
Last ObjectModification:
2017_10_13-PM-06_55_12
Theory : PZF
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