Nuprl Lemma : MultiTree-ext
∀[T:Type]
MultiTree(T) ≡ lbl:Atom × if lbl =a "Node" then labels:{L:Atom List| 0 < ||L||} × ({a:Atom| (a ∈ labels)} ⟶ MultiTr\000Cee(T))
if lbl =a "Leaf" then T
else Void
fi
Proof
Definitions occuring in Statement :
MultiTree: MultiTree(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],
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
member: t ∈ T,
MultiTree: MultiTree(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
guard: {T},
eq_atom: x =a y,
MultiTreeco_size: MultiTreeco_size(p),
pi1: fst(t),
pi2: snd(t),
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
has-value: (a)↓,
so_lambda: λ2x.t[x],
prop: ℙ,
int_seg: {i..j-},
nat: ℕ,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
MultiTree_size: MultiTree_size(p),
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
partial_wf,
value-type_wf,
squash_wf,
and_wf,
lelt_wf,
MultiTree_size_wf,
sum-nat,
false_wf,
add-nat,
set_wf,
MultiTreeco_wf,
subtype_rel_dep_function,
list_wf,
ifthenelse_wf,
neg_assert_of_eq_atom,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
MultiTree_wf,
set-value-type,
has-value_wf-partial,
sum-partial-has-value,
equal-wf-base-T,
less_than_wf,
int-value-type,
value-type-has-value,
int_subtype_base,
le_wf,
set_subtype_base,
nat_wf,
subtype_partial_sqtype_base,
length_wf,
int_seg_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
int_seg_properties,
list-subtype,
l_member_wf,
select_wf,
MultiTreeco_size_wf,
length_wf_nat,
sum-partial-nat,
atom_subtype_base,
subtype_base_sq,
assert_of_eq_atom,
eqtt_to_assert,
bool_wf,
eq_atom_wf,
MultiTreeco-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
lemma_by_obid,
hypothesis,
isectElimination,
hypothesisEquality,
promote_hyp,
productElimination,
hypothesis_subsumption,
applyEquality,
sqequalRule,
dependent_pairEquality,
tokenEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
because_Cache,
instantiate,
cumulativity,
atomEquality,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
callbyvalueAdd,
baseClosed,
setEquality,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination,
equalityEquality,
dependent_set_memberEquality,
productEquality,
functionEquality,
universeEquality,
sqleReflexivity,
imageMemberEquality,
introduction,
substitution
Latex:
\mforall{}[T:Type]
MultiTree(T) \mequiv{} lbl:Atom \mtimes{} if lbl =a "Node"
then labels:\{L:Atom List| 0 < ||L||\}
\mtimes{} (\{a:Atom| (a \mmember{} labels)\} {}\mrightarrow{} MultiTree(T))
if lbl =a "Leaf" then T
else Void
fi
Date html generated:
2016_05_16-AM-08_52_44
Last ObjectModification:
2016_01_17-AM-09_42_53
Theory : C-semantics
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