Nuprl Lemma : u-almost-full-finite-intersection
∀k:ℕ. ∀A:ℕk ⟶ ℕ ⟶ ℙ. ((∀i:ℕk. u-almost-full(n.A[i;n])) ⇒ u-almost-full(n.∀i:ℕk. A[i;n]))
Proof
Definitions occuring in Statement :
u-almost-full: u-almost-full(n.A[n]),
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
and: P ∧ Q,
guard: {T},
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
label: ...$L... t,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
all_wf,
int_seg_wf,
u-almost-full_wf,
nat_wf,
subtract_wf,
set_wf,
less_than_wf,
primrec-wf2,
u-almost-full-filter,
true_wf,
int_seg_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
subtype_rel_self,
decidable__lt,
lelt_wf,
decidable__equal_int,
equal_wf,
squash_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
functionEquality,
cumulativity,
universeEquality,
rename,
setElimination,
instantiate,
because_Cache,
intEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
dependent_set_memberEquality,
productEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed
Latex:
\mforall{}k:\mBbbN{}. \mforall{}A:\mBbbN{}k {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}. ((\mforall{}i:\mBbbN{}k. u-almost-full(n.A[i;n])) {}\mRightarrow{} u-almost-full(n.\mforall{}i:\mBbbN{}k. A[i;n]))
Date html generated:
2017_04_20-AM-07_24_14
Last ObjectModification:
2017_02_27-PM-05_59_18
Theory : continuity
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