Nuprl Lemma : bs_tree_lookup_wf
∀[E:Type]. ∀[cmp:comparison(E)]. ∀[x:E]. ∀[tr:ordered_bs_tree(E;cmp)].
(bs_tree_lookup(cmp;x;tr) ∈ (∃z:E [(((cmp z x) = 0 ∈ ℤ) ∧ z ∈ tr)]) ∨ (↓∀z:E. (z ∈ tr ⇒ (¬((cmp z x) = 0 ∈ ℤ)))))
Proof
Definitions occuring in Statement :
bs_tree_lookup: bs_tree_lookup(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],
sq_exists: ∃x:A [B[x]],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ordered_bs_tree: ordered_bs_tree(E;cmp),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
comparison: comparison(T),
so_apply: x[s],
guard: {T},
member_bs_tree: x ∈ tr,
bs_tree_lookup: bs_tree_lookup(cmp;x;tr),
bs_tree_ind: bs_tree_ind,
bst_null: bst_null(),
false: False,
bst_leaf: bst_leaf(value),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
cand: A c∧ B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
not: ¬A,
nequal: a ≠ b ∈ T ,
bs_tree_ordered: bs_tree_ordered(E;cmp;tr),
bst_node: bst_node(left;value;right),
has-value: (a)↓,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
sq_exists: ∃x:A [B[x]]
Lemmas referenced :
ordered_bs_tree_wf,
comparison_wf,
bs_tree-induction,
bs_tree_ordered_wf,
bs_tree_lookup_wf1,
unit_wf2,
equal-wf-T-base,
member_bs_tree_wf,
all_wf,
not_wf,
equal_wf,
bs_tree_wf,
false_wf,
bst_null_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
and_wf,
bst_leaf_wf,
bst_node_wf,
value-type-has-value,
int-value-type,
lt_int_wf,
assert_of_lt_int,
top_wf,
less_than_wf,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
or_wf,
le_wf,
itermMinus_wf,
int_term_value_minus_lemma,
squash_wf,
true_wf,
comparison-anti,
subtype_rel_self,
iff_weakening_equal,
int_subtype_base,
sq_exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
hypothesisEquality,
isect_memberEquality,
because_Cache,
dependent_functionElimination,
universeEquality,
lambdaEquality,
functionEquality,
unionEquality,
lambdaFormation,
unionElimination,
productEquality,
intEquality,
applyEquality,
baseClosed,
independent_functionElimination,
voidElimination,
natural_numberEquality,
equalityElimination,
productElimination,
independent_isectElimination,
independent_pairFormation,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
dependent_set_memberEquality,
applyLambdaEquality,
callbyvalueReduce,
lessCases,
axiomSqEquality,
voidEquality,
imageMemberEquality,
imageElimination,
approximateComputation,
int_eqEquality,
inlFormation,
inrFormation,
hyp_replacement,
equalityUniverse,
levelHypothesis,
minusEquality,
inlEquality,
inrEquality
Latex:
\mforall{}[E:Type]. \mforall{}[cmp:comparison(E)]. \mforall{}[x:E]. \mforall{}[tr:ordered\_bs\_tree(E;cmp)].
(bs\_tree\_lookup(cmp;x;tr) \mmember{} (\mexists{}z:E [(((cmp z x) = 0) \mwedge{} z \mmember{} tr)])
\mvee{} (\mdownarrow{}\mforall{}z:E. (z \mmember{} tr {}\mRightarrow{} (\mneg{}((cmp z x) = 0)))))
Date html generated:
2019_10_15-AM-10_47_54
Last ObjectModification:
2018_08_21-PM-01_58_46
Theory : tree_1
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