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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