Nuprl Lemma : isl-prior-iff
∀[T:Type]. ∀f:ℕ ⟶ (T + Top). ∀n:ℕ. (↑isl(prior(n;f)) ⇐⇒ ∃k:ℕn. (↑isl(f k)))
Proof
Definitions occuring in Statement :
prior: prior(n;f),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
prop: ℙ,
and: P ∧ Q,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
true: True,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
prior-cases,
prior_wf,
nat_wf,
int_seg_wf,
unit_wf2,
equal_wf,
top_wf,
and_wf,
isl_wf,
btrue_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert_wf,
int_seg_subtype_nat,
false_wf,
true_wf,
exists_wf,
all_wf,
not_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
cumulativity,
functionExtensionality,
applyEquality,
unionEquality,
productEquality,
natural_numberEquality,
setElimination,
rename,
unionElimination,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionEquality,
universeEquality,
productElimination,
independent_pairFormation,
dependent_pairFormation,
dependent_set_memberEquality,
applyLambdaEquality,
instantiate,
independent_isectElimination,
lambdaEquality,
because_Cache,
inlEquality,
addEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}f:\mBbbN{} {}\mrightarrow{} (T + Top). \mforall{}n:\mBbbN{}. (\muparrow{}isl(prior(n;f)) \mLeftarrow{}{}\mRightarrow{} \mexists{}k:\mBbbN{}n. (\muparrow{}isl(f k)))
Date html generated:
2017_10_01-AM-09_12_14
Last ObjectModification:
2017_07_26-PM-04_47_58
Theory : general
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