Nuprl Lemma : MultiTreeco-ext
∀[T:Type]
MultiTreeco(T) ≡ lbl:Atom × if lbl =a "Node" then labels:{L:Atom List| 0 < ||L||} × ({a:Atom| (a ∈ labels)} ⟶ Multi\000CTreeco(T))
if lbl =a "Leaf" then T
else Void
fi
Proof
Definitions occuring in Statement :
MultiTreeco: MultiTreeco(T),
l_member: (x ∈ l),
length: ||as||,
list: T List,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
ext-eq: A ≡ B,
less_than: a < b,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
token: "$token",
atom: Atom,
void: Void,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
MultiTreeco: MultiTreeco(T),
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
continuous-monotone: ContinuousMonotone(T.F[T]),
and: P ∧ Q,
type-monotone: Monotone(T.F[T]),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
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,
ifthenelse_wf,
eq_atom_wf,
list_wf,
less_than_wf,
length_wf,
l_member_wf,
subtype_rel_product,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_rel_dep_function,
set_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
subtype_rel_wf,
strong-continuous-depproduct,
strong-continuous-function,
continuous-id,
continuous-constant,
subtype_rel_weakening,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
productEquality,
atomEquality,
instantiate,
hypothesisEquality,
tokenEquality,
hypothesis,
universeEquality,
setEquality,
natural_numberEquality,
functionEquality,
setElimination,
rename,
voidEquality,
independent_isectElimination,
independent_pairFormation,
because_Cache,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
cumulativity,
equalityEquality,
axiomEquality,
isect_memberEquality,
isectEquality,
applyEquality,
independent_pairEquality
Latex:
\mforall{}[T:Type]
MultiTreeco(T) \mequiv{} lbl:Atom \mtimes{} if lbl =a "Node"
then labels:\{L:Atom List| 0 < ||L||\}
\mtimes{} (\{a:Atom| (a \mmember{} labels)\} {}\mrightarrow{} MultiTreeco(T))
if lbl =a "Leaf" then T
else Void
fi
Date html generated:
2016_05_16-AM-08_52_21
Last ObjectModification:
2015_12_28-PM-06_54_43
Theory : C-semantics
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