Nuprl Lemma : bs_tree_lookup_wf1
∀[E:Type]. ∀[cmp:comparison(E)]. ∀[x:E]. ∀[tr:bs_tree(E)]. (bs_tree_lookup(cmp;x;tr) ∈ E?)
Proof
Definitions occuring in Statement :
bs_tree_lookup: bs_tree_lookup(cmp;x;tr),
bs_tree: bs_tree(E),
comparison: comparison(T),
uall: ∀[x:A]. B[x],
unit: Unit,
member: t ∈ T,
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bs_tree_lookup: bs_tree_lookup(cmp;x;tr),
so_lambda: λ2x.t[x],
comparison: comparison(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
so_apply: x[s],
so_lambda: so_lambda(x,y,z,w,v.t[x; y; z; w; v]),
has-value: (a)↓,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
top: Top,
not: ¬A,
so_apply: x[s1;s2;s3;s4;s5]
Lemmas referenced :
bs_tree_ind_wf_simple,
unit_wf2,
it_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
value-type-has-value,
int-value-type,
top_wf,
less_than_wf,
bs_tree_wf,
comparison_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
unionEquality,
because_Cache,
hypothesis,
inrEquality,
lambdaEquality,
applyEquality,
setElimination,
rename,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
inlEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
callbyvalueReduce,
intEquality,
lessCases,
independent_pairFormation,
baseClosed,
imageMemberEquality,
axiomSqEquality,
isect_memberEquality,
voidEquality,
imageElimination,
axiomEquality,
universeEquality
Latex:
\mforall{}[E:Type]. \mforall{}[cmp:comparison(E)]. \mforall{}[x:E]. \mforall{}[tr:bs\_tree(E)]. (bs\_tree\_lookup(cmp;x;tr) \mmember{} E?)
Date html generated:
2019_10_15-AM-10_47_45
Last ObjectModification:
2018_08_20-PM-09_41_27
Theory : tree_1
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