Nuprl Lemma : equal-nat-inf-infinity
∀[x:ℕ∞]. uiff(x = ∞ ∈ ℕ∞;∀i:ℕ. (¬(x = i∞ ∈ ℕ∞)))
Proof
Definitions occuring in Statement :
nat-inf-infinity: ∞,
nat2inf: n∞,
nat-inf: ℕ∞,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat-inf: ℕ∞,
nat-inf-infinity: ∞,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
less_than: a < b,
nat2inf: n∞,
assert: ↑b,
ifthenelse: if b then t else f fi ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
sq_type: SQType(T),
bnot: ¬bb,
subtract: n - m,
squash: ↓T
Lemmas referenced :
equal_wf,
nat-inf_wf,
nat2inf_wf,
nat_wf,
equal-wf-T-base,
all_wf,
not_wf,
nat-inf-infinity-new,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
bool_wf,
decidable__lt,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
assert_wf,
iff_imp_equal_bool,
lt_int_wf,
true_wf,
assert_of_lt_int,
iff_wf,
add-zero,
decidable__equal_bool,
assert_elim,
btrue_neq_bfalse,
subtype_base_sq,
int_subtype_base,
eqtt_to_assert,
equal-wf-base,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
minus-one-mul,
add-commutes,
add-associates,
add-mul-special,
zero-mul,
zero-add,
squash_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
because_Cache,
baseClosed,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
hyp_replacement,
applyLambdaEquality,
setElimination,
rename,
dependent_set_memberEquality,
functionExtensionality,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
unionElimination,
applyEquality,
hypothesis_subsumption,
equalityElimination,
addEquality,
functionEquality,
addLevel,
impliesFunctionality,
instantiate,
cumulativity,
promote_hyp,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
imageMemberEquality
Latex:
\mforall{}[x:\mBbbN{}\minfty{}]. uiff(x = \minfty{};\mforall{}i:\mBbbN{}. (\mneg{}(x = i\minfty{})))
Date html generated:
2017_10_01-AM-08_29_25
Last ObjectModification:
2017_07_26-PM-04_23_57
Theory : basic
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