Nuprl Lemma : not-LPO
¬(∀f:ℕ ⟶ ℕ. ((∀n:ℕ. ((f n) = 0 ∈ ℤ)) ∨ (¬(∀n:ℕ. ((f n) = 0 ∈ ℤ)))))
Proof
Definitions occuring in Statement :
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
less_than: a < b,
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
false_wf,
le_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
nat_wf,
less_than_wf,
all_wf,
equal-wf-T-base,
iff_wf,
or_wf,
not_wf,
exists_wf,
weak-continuity-nat-nat,
squash-from-quotient,
equal-wf-base-T,
subtype_rel_dep_function,
int_seg_wf,
int_seg_subtype_nat,
subtype_rel_self,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
int_seg_properties,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_subtype_base,
decidable__lt,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
unionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
intEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
because_Cache,
functionExtensionality,
setElimination,
rename,
voidElimination,
functionEquality,
promote_hyp,
equalityElimination,
instantiate,
cumulativity,
approximateComputation,
isect_memberEquality,
voidEquality,
int_eqEquality,
applyLambdaEquality,
impliesFunctionality
Latex:
\mneg{}(\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((\mforall{}n:\mBbbN{}. ((f n) = 0)) \mvee{} (\mneg{}(\mforall{}n:\mBbbN{}. ((f n) = 0)))))
Date html generated:
2018_05_21-PM-01_17_55
Last ObjectModification:
2017_10_14-PM-09_14_13
Theory : continuity
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