Nuprl Lemma : nat-inf-limit
∀p:ℕ∞ ⟶ 𝔹. ((∀n:ℕ. p n∞ = ff) ⇒ p ∞ = ff)
Proof
Definitions occuring in Statement :
nat-inf-infinity: ∞,
nat2inf: n∞,
nat-inf: ℕ∞,
nat: ℕ,
bfalse: ff,
bool: 𝔹,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nat-inf: ℕ∞,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
nat2inf: n∞,
nat-inf-infinity: ∞
Lemmas referenced :
nat-inf-infinity_wf,
eqtt_to_assert,
no-weak-limited-omniscience,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
bfalse_wf,
istype-nat,
nat2inf_wf,
nat-inf_wf,
bnot_wf,
b-exists_wf,
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,
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,
int_seg_wf,
assert_of_bnot,
assert-b-exists,
istype-assert,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
decidable__equal_int,
subtract_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
subtype_rel_self,
int_seg_subtype_nat,
istype-false,
btrue_wf,
equal-wf-T-base,
nat_wf,
le_wf,
primrec-wf2,
decidable__exists_int_seg,
decidable__equal_bool,
easy-member-int_seg,
add-associates,
add-swap,
add-commutes,
zero-add,
iff_imp_equal_bool,
lt_int_wf,
istype-void,
iff_weakening_uiff,
assert_wf,
not_wf,
less_than_wf,
assert_of_lt_int,
istype-true,
assert_elim,
btrue_neq_bfalse,
assert_functionality_wrt_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
applyEquality,
hypothesisEquality,
introduction,
extract_by_obid,
hypothesis,
inhabitedIsType,
thin,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
independent_functionElimination,
voidElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
because_Cache,
sqequalRule,
functionIsType,
universeIsType,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
addEquality,
setElimination,
rename,
natural_numberEquality,
approximateComputation,
int_eqEquality,
Error :memTop,
independent_pairFormation,
imageElimination,
productIsType,
intEquality,
applyLambdaEquality,
hypothesis_subsumption,
functionEquality,
productEquality,
baseClosed,
baseApply,
closedConclusion,
setIsType,
imageMemberEquality,
sqequalBase,
inlFormation_alt,
inrFormation_alt,
hyp_replacement
Latex:
\mforall{}p:\mBbbN{}\minfty{} {}\mrightarrow{} \mBbbB{}. ((\mforall{}n:\mBbbN{}. p n\minfty{} = ff) {}\mRightarrow{} p \minfty{} = ff)
Date html generated:
2020_05_20-AM-07_47_51
Last ObjectModification:
2020_02_28-PM-02_49_04
Theory : basic
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