Nuprl Lemma : nat-inf-attach-unit
∀F:ℕ ⟶ Type. ∃G:ℕ∞ ⟶ Type. ((∀n:ℕ. G n∞ ~ F n) ∧ G ∞ ~ ℕ1)
Proof
Definitions occuring in Statement : 
nat-inf-infinity: ∞
, 
nat2inf: n∞
, 
nat-inf: ℕ∞
, 
equipollent: A ~ B
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
and: P ∧ Q
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
natural_number: $n
, 
universe: Type
Definitions unfolded in proof : 
top: Top
, 
false: False
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
not: ¬A
, 
or: P ∨ Q
, 
decidable: Dec(P)
, 
ge: i ≥ j 
, 
guard: {T}
, 
sq_type: SQType(T)
, 
nat: ℕ
, 
pi1: fst(t)
, 
uimplies: b supposing a
, 
subtype_rel: A ⊆r B
, 
implies: P 
⇒ Q
, 
singleton-type: singleton-type(A)
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
Lemmas referenced : 
nat-inf-infinity-new, 
equal-wf-base-T, 
equipollent-empty-domain, 
int_formula_prop_wf, 
int_term_value_var_lemma, 
int_formula_prop_eq_lemma, 
int_formula_prop_not_lemma, 
itermVar_wf, 
intformeq_wf, 
intformnot_wf, 
full-omega-unsat, 
decidable__equal_int, 
nat_properties, 
le_wf, 
set_subtype_base, 
int_subtype_base, 
subtype_base_sq, 
subtype_rel_self, 
subtype_rel_dep_function, 
equipollent-singleton-domain, 
set_wf, 
nat2inf-one-one, 
nat_wf, 
equal_wf, 
nat-inf_wf, 
nat2inf_wf, 
all_wf, 
equipollent_wf, 
nat-inf-infinity_wf, 
int_seg_wf
Rules used in proof : 
baseClosed, 
voidEquality, 
voidElimination, 
isect_memberEquality, 
int_eqEquality, 
approximateComputation, 
unionElimination, 
equalityTransitivity, 
productElimination, 
intEquality, 
independent_isectElimination, 
instantiate, 
dependent_functionElimination, 
equalitySymmetry, 
independent_functionElimination, 
because_Cache, 
dependent_set_memberEquality, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
dependent_pairFormation, 
lambdaEquality, 
functionEquality, 
setEquality, 
cut, 
introduction, 
extract_by_obid, 
hypothesis, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
applyEquality, 
functionExtensionality, 
setElimination, 
rename, 
independent_pairFormation, 
productEquality, 
sqequalRule, 
natural_numberEquality, 
cumulativity, 
universeEquality
Latex:
\mforall{}F:\mBbbN{}  {}\mrightarrow{}  Type.  \mexists{}G:\mBbbN{}\minfty{}  {}\mrightarrow{}  Type.  ((\mforall{}n:\mBbbN{}.  G  n\minfty{}  \msim{}  F  n)  \mwedge{}  G  \minfty{}  \msim{}  \mBbbN{}1)
Date html generated:
2018_07_29-AM-09_29_18
Last ObjectModification:
2018_07_27-PM-05_28_45
Theory : basic
Home
Index