Nuprl Lemma : bs_tree-ext
∀[E:Type]
bs_tree(E) ≡ lbl:Atom × if lbl =a "null" then Unit
if lbl =a "leaf" then E
if lbl =a "node" then left:bs_tree(E) × value:E × bs_tree(E)
else Void
fi
Proof
Definitions occuring in Statement :
bs_tree: bs_tree(E),
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
unit: Unit,
product: x:A × B[x],
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,
bs_tree: bs_tree(E),
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,
bs_treeco_size: bs_treeco_size(p),
has-value: (a)↓,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False,
pi1: fst(t),
pi2: snd(t),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bs_tree_size: bs_tree_size(p),
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A
Lemmas referenced :
bs_treeco-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
unit_wf2,
unit_subtype_base,
it_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
int_subtype_base,
bs_treeco_size_wf,
subtype_partial_sqtype_base,
nat_wf,
set_subtype_base,
le_wf,
base_wf,
value-type-has-value,
int-value-type,
has-value_wf-partial,
set-value-type,
bs_tree_wf,
ifthenelse_wf,
bs_treeco_wf,
add-nat,
false_wf,
bs_tree_size_wf,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
introduction,
extract_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,
dependent_pairFormation,
voidElimination,
dependent_set_memberEquality,
natural_numberEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
callbyvalueAdd,
productEquality,
universeEquality,
voidEquality,
sqleReflexivity
Latex:
\mforall{}[E:Type]
bs\_tree(E) \mequiv{} lbl:Atom \mtimes{} if lbl =a "null" then Unit
if lbl =a "leaf" then E
if lbl =a "node" then left:bs\_tree(E) \mtimes{} value:E \mtimes{} bs\_tree(E)
else Void
fi
Date html generated:
2017_10_01-AM-08_30_45
Last ObjectModification:
2017_07_26-PM-04_24_44
Theory : tree_1
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