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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