Nuprl Lemma : generic-countable-intersection
∀[T:Type]. ∀[S:ℕ ⟶ (ℕ ⟶ T) ⟶ ℙ']. ((∀i:ℕ. Generic{x:ℕ⟶T|S[i;x]}) ⇒ Generic{x:ℕ⟶T|∀i:ℕ. S[i;x]})
Proof
Definitions occuring in Statement :
generic: Generic{f:ℕ⟶T|S[f]},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
generic: Generic{f:ℕ⟶T|S[f]},
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
nat: ℕ,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
cand: A c∧ B,
pi1: fst(t),
surject: Surj(A;B;f),
sq_type: SQType(T)
Lemmas referenced :
all_wf,
nat_wf,
generic_wf,
pair-coding-exists,
exists_wf,
list_wf,
iseg_wf,
int_seg_wf,
length_wf,
equal_wf,
select_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_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_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_subtype_nat,
false_wf,
pi1_wf_top,
subtype_base_sq,
product_subtype_base,
set_subtype_base,
le_wf,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesis,
lambdaEquality,
hypothesisEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
rename,
productElimination,
dependent_pairFormation,
because_Cache,
productEquality,
natural_numberEquality,
setElimination,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
promote_hyp
Latex:
\mforall{}[T:Type]. \mforall{}[S:\mBbbN{} {}\mrightarrow{} (\mBbbN{} {}\mrightarrow{} T) {}\mrightarrow{} \mBbbP{}'].
((\mforall{}i:\mBbbN{}. Generic\{x:\mBbbN{}{}\mrightarrow{}T|S[i;x]\}) {}\mRightarrow{} Generic\{x:\mBbbN{}{}\mrightarrow{}T|\mforall{}i:\mBbbN{}. S[i;x]\})
Date html generated:
2018_05_21-PM-07_57_15
Last ObjectModification:
2017_07_26-PM-05_34_47
Theory : general
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