Nuprl Lemma : gamma-neighbourhood-prop6
∀beta:ℕ ⟶ ℕ. ∀n0:finite-nat-seq(). ∀x,n:ℕ.
((¬((beta x) = 0 ∈ ℤ))
⇒ (↑isl(gamma-neighbourhood(beta;n0) ext-finite-nat-seq(n0**λk.(x + 1)^(1);0)^(n)))
⇒ ((gamma-neighbourhood(beta;n0) ext-finite-nat-seq(n0**λk.(x + 1)^(1);0)^(n)) = (inl 0) ∈ (ℕ?)))
Proof
Definitions occuring in Statement :
ext-finite-nat-seq: ext-finite-nat-seq(f;x),
gamma-neighbourhood: gamma-neighbourhood(beta;n0),
append-finite-nat-seq: f**g,
mk-finite-nat-seq: f^(n),
finite-nat-seq: finite-nat-seq(),
nat: ℕ,
assert: ↑b,
isl: isl(x),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
unit: Unit,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
subtype_rel: A ⊆r B,
isl: isl(x),
gamma-neighbourhood: gamma-neighbourhood(beta;n0),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
assert: ↑b,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
finite-nat-seq: finite-nat-seq(),
mk-finite-nat-seq: f^(n),
append-finite-nat-seq: f**g,
ext-finite-nat-seq: ext-finite-nat-seq(f;x),
pi1: fst(t),
pi2: snd(t),
less_than: a < b,
true: True,
squash: ↓T,
rev_implies: P ⇐ Q
Lemmas referenced :
istype-assert,
gamma-neighbourhood_wf,
mk-finite-nat-seq_wf,
ext-finite-nat-seq_wf,
append-finite-nat-seq_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
int_seg_properties,
nat_properties,
intformand_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_seg_wf,
btrue_wf,
bfalse_wf,
istype-nat,
finite-nat-seq_wf,
init-seg-nat-seq_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
extend-seq1-all-dec,
subtype_rel_self,
nat_wf,
decidable_wf,
assert_wf,
not_wf,
equal-wf-base,
set_subtype_base,
le_wf,
int_subtype_base,
int_seg_subtype_nat,
istype-false,
subtype_rel_function,
all_wf,
exists_wf,
true_wf,
decidable__lt,
intformless_wf,
intformeq_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
istype-less_than,
assert-init-seg-nat-seq2,
istype-top,
less_than_anti-reflexive,
less_than_wf,
lt_int_wf,
assert_of_lt_int,
decidable__equal_int,
iff_weakening_uiff,
unit_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
because_Cache,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
Error :universeIsType,
addEquality,
setElimination,
rename,
productElimination,
int_eqEquality,
independent_pairFormation,
Error :inhabitedIsType,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
Error :functionIsType,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
functionEquality,
productEquality,
intEquality,
baseClosed,
closedConclusion,
Error :productIsType,
hyp_replacement,
applyLambdaEquality,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isectIsTypeImplies,
imageMemberEquality,
imageElimination,
baseApply,
sqequalBase,
Error :inlEquality_alt
Latex:
\mforall{}beta:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}n0:finite-nat-seq(). \mforall{}x,n:\mBbbN{}.
((\mneg{}((beta x) = 0))
{}\mRightarrow{} (\muparrow{}isl(gamma-neighbourhood(beta;n0) ext-finite-nat-seq(n0**\mlambda{}k.(x + 1)\^{}(1);0)\^{}(n)))
{}\mRightarrow{} ((gamma-neighbourhood(beta;n0) ext-finite-nat-seq(n0**\mlambda{}k.(x + 1)\^{}(1);0)\^{}(n)) = (inl 0)))
Date html generated:
2019_06_20-PM-03_04_43
Last ObjectModification:
2018_12_06-PM-11_34_43
Theory : continuity
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