Nuprl Lemma : extend-seq1-all-dec
∀n,n0:finite-nat-seq(). ∀beta:ℕ ⟶ ℕ.
Dec(∃x:ℕ. ((↑init-seg-nat-seq(n0**λi.x^(1);n)) ∧ (¬((beta x) = 0 ∈ ℤ)) ∧ (∀y:ℕx. ((beta y) = 0 ∈ ℤ))))
Proof
Definitions occuring in Statement :
init-seg-nat-seq: init-seg-nat-seq(f;g),
append-finite-nat-seq: f**g,
mk-finite-nat-seq: f^(n),
finite-nat-seq: finite-nat-seq(),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
decidable: Dec(P),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
exists: ∃x:A. B[x],
and: P ∧ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
int_seg: {i..j-},
lelt: i ≤ j < k,
finite-nat-seq: finite-nat-seq(),
decidable: Dec(P),
or: P ∨ Q,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
pi2: snd(t),
pi1: fst(t),
iff: P ⇐⇒ Q,
append-finite-nat-seq: f**g,
mk-finite-nat-seq: f^(n),
guard: {T},
sq_type: SQType(T),
squash: ↓T,
true: True,
top: Top,
less_than: a < b,
cand: A c∧ B,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
init-seg-nat-seq: init-seg-nat-seq(f;g)
Lemmas referenced :
nat_wf,
assert_wf,
init-seg-nat-seq_wf,
append-finite-nat-seq_wf,
mk-finite-nat-seq_wf,
istype-void,
istype-le,
int_seg_wf,
not_wf,
equal-wf-base,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
int_seg_subtype_nat,
istype-false,
decidable__le,
istype-nat,
finite-nat-seq_wf,
decidable__assert,
decidable__equal_int,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
istype-less_than,
false_wf,
assert-init-seg-nat-seq2,
subtype_base_sq,
lelt_wf,
equal_wf,
less_than_wf,
less_than_anti-reflexive,
top_wf,
decidable__all_int_seg,
int_seg_properties,
istype-assert,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__equal_nat,
assert-init-seg-nat-seq,
append-finite-nat-seq-assoc,
ble_wf,
assert-ble
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
productEquality,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
voidElimination,
lambdaEquality_alt,
universeIsType,
intEquality,
applyEquality,
independent_isectElimination,
baseClosed,
functionEquality,
setElimination,
rename,
because_Cache,
productElimination,
dependent_functionElimination,
addEquality,
unionElimination,
functionIsType,
inhabitedIsType,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
productIsType,
equalitySymmetry,
equalityTransitivity,
lambdaFormation,
inrFormation,
lambdaEquality,
dependent_set_memberEquality,
dependent_pairEquality,
cumulativity,
instantiate,
dependent_pairFormation,
functionExtensionality,
applyLambdaEquality,
hyp_replacement,
imageElimination,
imageMemberEquality,
voidEquality,
isect_memberEquality,
axiomSqEquality,
isect_memberFormation,
lessCases,
inlFormation_alt,
dependent_pairEquality_alt,
equalityIstype,
sqequalBase,
functionExtensionality_alt,
equalityElimination,
isect_memberFormation_alt,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality,
promote_hyp
Latex:
\mforall{}n,n0:finite-nat-seq(). \mforall{}beta:\mBbbN{} {}\mrightarrow{} \mBbbN{}.
Dec(\mexists{}x:\mBbbN{}. ((\muparrow{}init-seg-nat-seq(n0**\mlambda{}i.x\^{}(1);n)) \mwedge{} (\mneg{}((beta x) = 0)) \mwedge{} (\mforall{}y:\mBbbN{}x. ((beta y) = 0))))
Date html generated:
2020_05_19-PM-10_05_58
Last ObjectModification:
2020_01_04-PM-08_03_49
Theory : continuity
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