Nuprl Lemma : unary-almost-full-has-strict-inc
∀A:ℕ ⟶ ℙ. ((∀s:StrictInc. ⇃(∃n:ℕ. A[s n])) 
⇒ ⇃(∃s:StrictInc. ∀n:ℕ. A[s n]))
Proof
Definitions occuring in Statement : 
strict-inc: StrictInc
, 
quotient: x,y:A//B[x; y]
, 
nat: ℕ
, 
prop: ℙ
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
implies: P 
⇒ Q
, 
true: True
, 
apply: f a
, 
function: x:A ⟶ B[x]
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
exists: ∃x:A. B[x]
, 
so_apply: x[s]
, 
strict-inc: StrictInc
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
uimplies: b supposing a
, 
prop: ℙ
, 
nat: ℕ
, 
ge: i ≥ j 
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
false: False
, 
top: Top
, 
and: P ∧ Q
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
le: A ≤ B
, 
less_than: a < b
, 
squash: ↓T
, 
guard: {T}
, 
cand: A c∧ B
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
sq_type: SQType(T)
, 
bnot: ¬bb
, 
assert: ↑b
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
subtract: n - m
, 
true: True
, 
so_lambda: λ2x.t[x]
Lemmas referenced : 
strict-inc_wf, 
quotient_wf, 
nat_wf, 
true_wf, 
istype-nat, 
equiv_rel_true, 
nat_properties, 
decidable__le, 
full-omega-unsat, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermAdd_wf, 
itermVar_wf, 
istype-int, 
int_formula_prop_and_lemma, 
istype-void, 
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, 
istype-le, 
int_seg_properties, 
decidable__lt, 
intformless_wf, 
int_formula_prop_less_lemma, 
int_seg_wf, 
istype-less_than, 
implies-quotient-true, 
less_than_wf, 
subtype_rel_self, 
axiom-choice-00-quot, 
implies-strict-inc, 
primrec_wf, 
primrec-unroll, 
lt_int_wf, 
eqtt_to_assert, 
assert_of_lt_int, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_wf, 
bool_subtype_base, 
assert-bnot, 
iff_weakening_uiff, 
assert_wf, 
add-associates, 
add-swap, 
add-commutes, 
zero-add, 
add-subtract-cancel, 
squash_wf, 
istype-universe, 
primrec0_lemma, 
subtract_wf, 
itermSubtract_wf, 
int_term_value_subtract_lemma, 
primrec-wf2
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :lambdaFormation_alt, 
cut, 
sqequalRule, 
Error :functionIsType, 
Error :universeIsType, 
introduction, 
extract_by_obid, 
hypothesis, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
productEquality, 
applyEquality, 
hypothesisEquality, 
setElimination, 
rename, 
because_Cache, 
Error :lambdaEquality_alt, 
Error :inhabitedIsType, 
Error :productIsType, 
independent_isectElimination, 
universeEquality, 
dependent_functionElimination, 
Error :dependent_set_memberEquality_alt, 
addEquality, 
natural_numberEquality, 
unionElimination, 
approximateComputation, 
independent_functionElimination, 
Error :dependent_pairFormation_alt, 
int_eqEquality, 
Error :isect_memberEquality_alt, 
voidElimination, 
independent_pairFormation, 
productElimination, 
imageElimination, 
instantiate, 
functionEquality, 
equalityElimination, 
equalityTransitivity, 
equalitySymmetry, 
Error :equalityIstype, 
promote_hyp, 
cumulativity, 
hyp_replacement, 
imageMemberEquality, 
baseClosed, 
applyLambdaEquality, 
Error :setIsType
Latex:
\mforall{}A:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}.  ((\mforall{}s:StrictInc.  \00D9(\mexists{}n:\mBbbN{}.  A[s  n]))  {}\mRightarrow{}  \00D9(\mexists{}s:StrictInc.  \mforall{}n:\mBbbN{}.  A[s  n]))
Date html generated:
2019_06_20-PM-02_57_32
Last ObjectModification:
2019_02_06-PM-03_58_56
Theory : continuity
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