Nuprl Lemma : sq_stable__member_bs_tree
∀[E:Type]. ∀cmp:comparison(E). ∀x:E. ∀tr:ordered_bs_tree(E;cmp).  SqStable(x ∈ tr)
Proof
Definitions occuring in Statement : 
ordered_bs_tree: ordered_bs_tree(E;cmp)
, 
member_bs_tree: x ∈ tr
, 
comparison: comparison(T)
, 
sq_stable: SqStable(P)
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
universe: Type
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
, 
so_apply: x[s]
, 
member_bs_tree: x ∈ tr
, 
bst_null: bst_null()
, 
bs_tree_ind: bs_tree_ind, 
false: False
, 
bst_leaf: bst_leaf(value)
, 
bst_node: bst_node(left;value;right)
, 
guard: {T}
, 
bs_tree_ordered: bs_tree_ordered(E;cmp;tr)
, 
and: P ∧ Q
, 
comparison: comparison(T)
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
true: True
, 
subtype_rel: A ⊆r B
, 
uimplies: b supposing a
, 
iff: P 
⇐⇒ Q
, 
less_than: a < b
, 
less_than': less_than'(a;b)
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
top: Top
Lemmas referenced : 
sq_stable__bs_tree_ordered, 
bs_tree-induction, 
bs_tree_ordered_wf, 
sq_stable_wf, 
member_bs_tree_wf, 
bs_tree_wf, 
sq_stable_from_decidable, 
false_wf, 
decidable__false, 
squash_wf, 
bst_null_wf, 
sq_stable__equal, 
equal_wf, 
bst_leaf_wf, 
bst_node_wf, 
ordered_bs_tree_wf, 
comparison_wf, 
or_wf, 
decidable__lt, 
less_than_wf, 
true_wf, 
comparison-reflexive, 
iff_weakening_equal, 
comparison-anti, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformless_wf, 
itermConstant_wf, 
itermMinus_wf, 
itermVar_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_less_lemma, 
int_term_value_constant_lemma, 
int_term_value_minus_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf
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, 
voidElimination, 
axiomEquality, 
universeEquality, 
productElimination, 
natural_numberEquality, 
applyEquality, 
unionElimination, 
equalitySymmetry, 
hyp_replacement, 
Error :applyLambdaEquality, 
equalityTransitivity, 
intEquality, 
independent_isectElimination, 
dependent_pairFormation, 
int_eqEquality, 
isect_memberEquality, 
voidEquality, 
independent_pairFormation, 
computeAll, 
inrFormation, 
inlFormation
Latex:
\mforall{}[E:Type].  \mforall{}cmp:comparison(E).  \mforall{}x:E.  \mforall{}tr:ordered\_bs\_tree(E;cmp).    SqStable(x  \mmember{}  tr)
Date html generated:
2016_10_25-AM-10_15_25
Last ObjectModification:
2016_07_12-AM-06_25_41
Theory : tree_1
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