Nuprl Lemma : unary-strong-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,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
so_apply: x[s],
strict-inc: StrictInc,
subtype_rel: A ⊆r B,
prop: ℙ,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
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,
cand: A c∧ B,
pi1: fst(t),
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),
guard: {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,
istype-nat,
subtype_rel_self,
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,
nat_wf,
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,
less_than_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
add-subtract-cancel,
squash_wf,
true_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 :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
introduction,
extract_by_obid,
hypothesis,
Error :productIsType,
applyEquality,
hypothesisEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
instantiate,
isectElimination,
universeEquality,
dependent_functionElimination,
Error :dependent_set_memberEquality_alt,
Error :lambdaEquality_alt,
addEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :inhabitedIsType,
productElimination,
imageElimination,
because_Cache,
promote_hyp,
functionExtensionality,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
cumulativity,
hyp_replacement,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
Error :setIsType
Latex:
\mforall{}[A:\mBbbN{} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}s:StrictInc. \mexists{}n:\mBbbN{}. A[s n]) {}\mRightarrow{} (\mexists{}s:StrictInc. \mforall{}n:\mBbbN{}. A[s n]))
Date html generated:
2019_06_20-PM-02_57_25
Last ObjectModification:
2019_02_06-PM-03_51_07
Theory : continuity
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