Nuprl Lemma : member-bs_tree_delete-implies
∀[E:Type]. ∀cmp:comparison(E). ∀x:E. ∀tr:bs_tree(E). ∀z:E. (z ∈ bs_tree_delete(cmp;x;tr) ⇒ z ∈ tr)
Proof
Definitions occuring in Statement :
bs_tree_delete: bs_tree_delete(cmp;x;tr),
member_bs_tree: x ∈ tr,
bs_tree: bs_tree(E),
comparison: comparison(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
guard: {T},
member_bs_tree: x ∈ tr,
bs_tree_delete: bs_tree_delete(cmp;x;tr),
bst_null: bst_null(),
bs_tree_ind: bs_tree_ind,
false: False,
bst_leaf: bst_leaf(value),
comparison: comparison(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
bst_node: bst_node(left;value;right),
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
pi2: snd(t),
pi1: fst(t)
Lemmas referenced :
bs_tree-induction,
all_wf,
member_bs_tree_wf,
bs_tree_delete_wf1,
bs_tree_wf,
comparison_wf,
false_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lt_int_wf,
assert_of_lt_int,
top_wf,
less_than_wf,
bst_null?_wf,
or_wf,
member-bs_tree_max,
bs_tree_max_wf1,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesisEquality,
functionEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
universeEquality,
voidElimination,
applyEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
lessCases,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
inrFormation,
inlFormation,
productEquality
Latex:
\mforall{}[E:Type]. \mforall{}cmp:comparison(E). \mforall{}x:E. \mforall{}tr:bs\_tree(E). \mforall{}z:E. (z \mmember{} bs\_tree\_delete(cmp;x;tr) {}\mRightarrow{} z \mmember{} tr)
Date html generated:
2017_10_01-AM-08_31_37
Last ObjectModification:
2017_07_26-PM-04_25_05
Theory : tree_1
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