Nuprl Lemma : nat-inf-attach
∀F:ℕ ⟶ Type. ∀T:Type. ∃G:ℕ∞ ⟶ Type. ((∀n:ℕ. G n∞ ~ F n) ∧ G ∞ ~ T)
Proof
Definitions occuring in Statement :
nat-inf-infinity: ∞,
nat2inf: n∞,
nat-inf: ℕ∞,
equipollent: A ~ B,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int-prod: Π(f[x] | x < k),
top: Top,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q
Lemmas referenced :
nat_wf,
int_seg_wf,
nat-inf-attach-unit,
ext-eq_weakening,
equipollent_weakening_ext-eq,
function_functionality_wrt_equipollent_left,
equipollent_functionality_wrt_equipollent,
equipollent-void-domain,
nat-inf-infinity_wf,
nat2inf_wf,
equipollent_wf,
all_wf,
and_wf,
nat-inf_wf,
equipollent-product,
primrec1_lemma,
le_wf,
false_wf,
int-prod_wf,
product_functionality_wrt_equipollent_right,
product_functionality_wrt_equipollent,
equipollent-multiply,
equipollent-product-one,
union_functionality_wrt_equipollent,
equipollent-product-zero,
equipollent-sum-zero
Rules used in proof :
multiplyEquality,
addLevel,
allFunctionality,
levelHypothesis,
andLevelFunctionality,
allLevelFunctionality,
unionEquality,
productEquality,
dependent_set_memberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
applyEquality,
hypothesisEquality,
independent_pairFormation,
because_Cache,
independent_functionElimination,
independent_isectElimination,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lambdaEquality,
isectElimination,
natural_numberEquality,
hypothesis,
productElimination,
sqequalRule,
rename,
universeEquality,
functionEquality,
cumulativity
Latex:
\mforall{}F:\mBbbN{} {}\mrightarrow{} Type. \mforall{}T:Type. \mexists{}G:\mBbbN{}\minfty{} {}\mrightarrow{} Type. ((\mforall{}n:\mBbbN{}. G n\minfty{} \msim{} F n) \mwedge{} G \minfty{} \msim{} T)
Date html generated:
2018_07_29-AM-09_29_23
Last ObjectModification:
2018_07_27-PM-05_34_13
Theory : basic
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