Nuprl Lemma : member-bs_tree_max
∀[E:Type]
∀tr:bs_tree(E). ∀d,z:E.
((z ∈ snd(bs_tree_max(tr;d)) ∨ (z = (fst(bs_tree_max(tr;d))) ∈ E)) ⇒ (z ∈ tr ∨ ((↑bst_null?(tr)) ∧ (z = d ∈ E))))
Proof
Definitions occuring in Statement :
bs_tree_max: bs_tree_max(tr;d),
member_bs_tree: x ∈ tr,
bst_null?: bst_null?(v),
bs_tree: bs_tree(E),
assert: ↑b,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
so_apply: x[s],
guard: {T},
member_bs_tree: x ∈ tr,
bs_tree_max: bs_tree_max(tr;d),
bst_null: bst_null(),
bs_tree_ind: bs_tree_ind,
bst_null?: bst_null?(v),
pi1: fst(t),
eq_atom: x =a y,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
pi2: snd(t),
false: False,
cand: A c∧ B,
true: True,
bst_leaf: bst_leaf(value),
bfalse: ff,
bst_node: bst_node(left;value;right),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
uimplies: b supposing a,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb
Lemmas referenced :
bs_tree-induction,
all_wf,
or_wf,
member_bs_tree_wf,
equal_wf,
assert_wf,
bst_null?_wf,
bs_tree_wf,
false_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bs_tree_max_wf1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
because_Cache,
functionEquality,
lambdaFormation,
equalityTransitivity,
hypothesis,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
productEquality,
universeEquality,
unionElimination,
voidElimination,
inrFormation,
natural_numberEquality,
independent_pairFormation,
inlFormation,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate
Latex:
\mforall{}[E:Type]
\mforall{}tr:bs\_tree(E). \mforall{}d,z:E.
((z \mmember{} snd(bs\_tree\_max(tr;d)) \mvee{} (z = (fst(bs\_tree\_max(tr;d)))))
{}\mRightarrow{} (z \mmember{} tr \mvee{} ((\muparrow{}bst\_null?(tr)) \mwedge{} (z = d))))
Date html generated:
2017_10_01-AM-08_31_19
Last ObjectModification:
2017_07_26-PM-04_25_01
Theory : tree_1
Home
Index