Nuprl Lemma : not-decidable-zero-sequence
¬(∀s:ℕ ⟶ ℕ. ((s = (λx.0) ∈ (ℕ ⟶ ℕ)) ∨ (¬(s = (λx.0) ∈ (ℕ ⟶ ℕ)))))
Proof
Definitions occuring in Statement :
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
half-squash-stable: half-squash-stable(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
less_than: a < b,
top: Top,
true: True,
squash: ↓T,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
nat_wf,
not_wf,
equal-wf-T-base,
strong-continuity2-implies-weak,
istype-false,
le_wf,
sq_stable-implies-half-squash-stable,
false_wf,
sq_stable_from_decidable,
decidable__false,
implies-quotient-true,
exists_wf,
all_wf,
equal-wf-base-T,
equal-wf-base,
int_seg_wf,
int_subtype_base,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
istype-void,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
int_seg_properties,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
set_subtype_base,
lelt_wf,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
subtype_rel_function,
int_seg_subtype_nat,
subtype_rel_self,
less_than_anti-reflexive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
rename,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
Error :inhabitedIsType,
hypothesisEquality,
Error :unionIsType,
Error :equalityIsType3,
thin,
baseClosed,
sqequalHypSubstitution,
isectElimination,
functionEquality,
dependent_functionElimination,
Error :lambdaEquality_alt,
because_Cache,
unionElimination,
Error :equalityIsType1,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
productElimination,
Error :productIsType,
Error :equalityIsType2,
setElimination,
Error :equalityIsType4,
applyEquality,
functionExtensionality,
voidElimination,
equalityElimination,
independent_isectElimination,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isect_memberEquality_alt,
imageMemberEquality,
imageElimination,
Error :dependent_pairFormation_alt,
baseApply,
closedConclusion,
promote_hyp,
instantiate,
cumulativity,
Error :functionExtensionality_alt,
approximateComputation,
intEquality,
int_eqEquality,
applyLambdaEquality
Latex:
\mneg{}(\mforall{}s:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((s = (\mlambda{}x.0)) \mvee{} (\mneg{}(s = (\mlambda{}x.0)))))
Date html generated:
2019_06_20-PM-02_56_57
Last ObjectModification:
2018_10_05-PM-08_21_08
Theory : continuity
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