Nuprl Lemma : member_bs_tree_insert
∀[E:Type]
∀cmp:comparison(E). ∀x:E. ∀tr:ordered_bs_tree(E;cmp). ∀y:E.
(y ∈ bs_tree_insert(cmp;x;tr) ⇐⇒ (y = x ∈ E) ∨ (y ∈ tr ∧ (¬((cmp x y) = 0 ∈ ℤ))))
Proof
Definitions occuring in Statement :
bs_tree_insert: bs_tree_insert(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,
or: P ∨ Q,
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,
comparison: comparison(T),
so_apply: x[s],
guard: {T},
member_bs_tree: x ∈ tr,
bs_tree_insert: bs_tree_insert(cmp;x;tr),
bst_null: bst_null(),
bs_tree_ind: bs_tree_ind,
bst_leaf: bst_leaf(value),
iff: P ⇐⇒ Q,
or: P ∨ Q,
false: False,
rev_implies: P ⇐ Q,
has-value: (a)↓,
uimplies: b supposing a,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
not: ¬A,
bst_node: bst_node(left;value;right),
cand: A c∧ B,
subtype_rel: A ⊆r B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
bs_tree_ordered: bs_tree_ordered(E;cmp;tr),
trans: Trans(T;x,y.E[x; y]),
decidable: Dec(P)
Lemmas referenced :
sq_stable__bs_tree_ordered,
bs_tree-induction,
bs_tree_ordered_wf,
all_wf,
iff_wf,
member_bs_tree_wf,
bs_tree_insert_wf1,
or_wf,
equal_wf,
not_wf,
equal-wf-T-base,
bs_tree_wf,
ordered_bs_tree_wf,
comparison_wf,
false_wf,
bst_null_wf,
value-type-has-value,
int-value-type,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
squash_wf,
true_wf,
comparison-anti,
iff_weakening_equal,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
itermMinus_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_term_value_minus_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bst_leaf_wf,
intformnot_wf,
int_formula_prop_not_lemma,
bst_node_wf,
strict-comparison-trans,
decidable__lt,
decidable__equal_int,
minus-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
functionEquality,
cumulativity,
productEquality,
intEquality,
applyEquality,
universeEquality,
independent_pairFormation,
inlFormation,
equalitySymmetry,
voidElimination,
unionElimination,
productElimination,
callbyvalueReduce,
independent_isectElimination,
natural_numberEquality,
equalityElimination,
equalityTransitivity,
lessCases,
sqequalAxiom,
isect_memberEquality,
voidEquality,
inrFormation,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
promote_hyp,
instantiate,
addLevel,
orFunctionality,
impliesFunctionality,
minusEquality,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[E:Type]
\mforall{}cmp:comparison(E). \mforall{}x:E. \mforall{}tr:ordered\_bs\_tree(E;cmp). \mforall{}y:E.
(y \mmember{} bs\_tree\_insert(cmp;x;tr) \mLeftarrow{}{}\mRightarrow{} (y = x) \mvee{} (y \mmember{} tr \mwedge{} (\mneg{}((cmp x y) = 0))))
Date html generated:
2017_10_01-AM-08_31_10
Last ObjectModification:
2017_07_26-PM-04_24_59
Theory : tree_1
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