Nuprl Lemma : enumerate_wf
∀[P:ℕ ⟶ 𝔹]. ∀[n:ℕ]. enumerate(P;n) ∈ {k:ℕ| ↑(P k)} supposing ∀n:ℕ. ∃k:ℕ. ((↑(P k)) ∧ (n ≤ k))
Proof
Definitions occuring in Statement :
enumerate: enumerate(P;n),
nat: ℕ,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
enumerate: enumerate(P;n),
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
subtract: n - m,
sq_type: SQType(T),
assert: ↑b,
true: True,
has-value: (a)↓
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
all_wf,
nat_wf,
exists_wf,
assert_wf,
le_wf,
primrec0_lemma,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
bool_wf,
false_wf,
mu-property,
primrec-unroll,
eq_int_wf,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
itermAdd_wf,
int_term_value_add_lemma,
add-is-int-iff,
set_subtype_base,
add-associates,
minus-add,
minus-one-mul,
add-swap,
add-mul-special,
add-commutes,
zero-add,
zero-mul,
add-zero,
subtype_base_sq,
assert_elim,
bool_subtype_base,
value-type-has-value,
int-value-type,
mu_wf,
add_nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
applyEquality,
functionExtensionality,
unionElimination,
because_Cache,
functionEquality,
dependent_set_memberEquality,
productElimination,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
impliesFunctionality,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
addEquality,
multiplyEquality,
instantiate,
cumulativity,
callbyvalueReduce
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbB{}]. \mforall{}[n:\mBbbN{}]. enumerate(P;n) \mmember{} \{k:\mBbbN{}| \muparrow{}(P k)\} supposing \mforall{}n:\mBbbN{}. \mexists{}k:\mBbbN{}. ((\muparrow{}(P k)) \mwedge{} (n \mleq{} k))
Date html generated:
2018_05_21-PM-07_58_35
Last ObjectModification:
2017_07_26-PM-05_35_51
Theory : general
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