Nuprl Lemma : member-bs_tree_delete
∀[E:Type]
∀cmp:comparison(E). ∀x:E. ∀tr:ordered_bs_tree(E;cmp). ∀z:E.
(z ∈ bs_tree_delete(cmp;x;tr) ⇐⇒ z ∈ tr ∧ (¬((cmp z x) = 0 ∈ ℤ)))
Proof
Definitions occuring in Statement :
bs_tree_delete: bs_tree_delete(cmp;x;tr),
ordered_bs_tree: ordered_bs_tree(E;cmp),
member_bs_tree: x ∈ tr,
comparison: comparison(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q,
apply: f a,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
ordered_bs_tree: ordered_bs_tree(E;cmp),
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
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,
not: ¬A,
bst_leaf: bst_leaf(value),
comparison: comparison(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
true: True,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
nequal: a ≠ b ∈ T ,
bs_tree_ordered: bs_tree_ordered(E;cmp;tr),
bst_node: bst_node(left;value;right),
less_than: a < b,
less_than': less_than'(a;b),
cand: A c∧ B,
trans: Trans(T;x,y.E[x; y]),
ext-eq: A ≡ B,
eq_atom: x =a y,
bst_null?: bst_null?(v),
pi1: fst(t)
Lemmas referenced :
sq_stable__bs_tree_ordered,
bs_tree-induction,
bs_tree_ordered_wf,
all_wf,
iff_wf,
member_bs_tree_wf,
bs_tree_delete_wf1,
not_wf,
equal-wf-base,
bs_tree_wf,
ordered_bs_tree_wf,
istype-universe,
comparison_wf,
istype-void,
bst_null_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
int_subtype_base,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
istype-int,
bst_leaf_wf,
equal_wf,
squash_wf,
true_wf,
comparison-anti,
subtype_rel_self,
iff_weakening_equal,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermMinus_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
minus-is-int-iff,
false_wf,
bst_node_wf,
bs_tree_max_wf,
sq_stable__iff,
sq_stable__member_bs_tree,
bs_tree_delete_wf,
sq_stable__and,
sq_stable__not,
lt_int_wf,
assert_of_lt_int,
istype-top,
iff_weakening_uiff,
assert_wf,
less_than_wf,
bst_null?_wf,
intformless_wf,
int_formula_prop_less_lemma,
strict-comparison-trans,
bs_tree-ext,
eq_atom_wf,
assert_of_eq_atom,
atom_subtype_base,
unit_wf2,
unit_subtype_base,
it_wf,
neg_assert_of_eq_atom,
le_wf,
bs_tree_max_wf1,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality_alt,
functionEquality,
because_Cache,
productEquality,
inhabitedIsType,
universeIsType,
universeEquality,
independent_pairFormation,
voidElimination,
productElimination,
productIsType,
independent_pairEquality,
functionIsTypeImplies,
applyEquality,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
intEquality,
equalityIsType1,
functionIsType,
equalityIsType4,
dependent_pairFormation_alt,
instantiate,
cumulativity,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
pointwiseFunctionality,
baseApply,
closedConclusion,
minusEquality,
dependent_set_memberEquality_alt,
lessCases,
axiomSqEquality,
inlFormation_alt,
unionIsType,
inrFormation_alt,
hypothesis_subsumption,
tokenEquality,
atomEquality,
equalityIsType2
Latex:
\mforall{}[E:Type]
\mforall{}cmp:comparison(E). \mforall{}x:E. \mforall{}tr:ordered\_bs\_tree(E;cmp). \mforall{}z:E.
(z \mmember{} bs\_tree\_delete(cmp;x;tr) \mLeftarrow{}{}\mRightarrow{} z \mmember{} tr \mwedge{} (\mneg{}((cmp z x) = 0)))
Date html generated:
2019_10_15-AM-10_47_43
Last ObjectModification:
2018_10_11-PM-08_56_55
Theory : tree_1
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