Nuprl Lemma : gamma-neighbourhood-prop4
∀beta:ℕ ⟶ ℕ. ∀n0:finite-nat-seq(). ∀x,n:ℕ.
((¬((beta x) = 0 ∈ ℤ))
⇒ (∀y:ℕx. ((beta y) = 0 ∈ ℤ))
⇒ (↑isl(gamma-neighbourhood(beta;n0) ext-finite-nat-seq(n0**λk.x^(1);0)^(n)))
⇒ ((gamma-neighbourhood(beta;n0) ext-finite-nat-seq(n0**λk.x^(1);0)^(n)) = (inl 1) ∈ (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(),
int_seg: {i..j-},
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,
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,
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;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,
cand: A c∧ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
finite-nat-seq: finite-nat-seq(),
pi1: fst(t),
append-finite-nat-seq: f**g,
init-seg-nat-seq: init-seg-nat-seq(f;g),
pi2: snd(t),
ge: i ≥ j ,
int_seg: {i..j-},
less_than: a < b,
true: True,
squash: ↓T,
lelt: i ≤ j < k,
mk-finite-nat-seq: f^(n),
ext-finite-nat-seq: ext-finite-nat-seq(f;x)
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_wf,
subtype_rel_function,
nat_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
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,
true_wf,
unit_wf2,
assert-init-seg-nat-seq2,
nat_properties,
intformand_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
lt_int_wf,
assert_of_lt_int,
int_seg_properties,
istype-less_than,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-top,
subtract_wf,
itermSubtract_wf,
intformless_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
decidable__lt,
int_seg_subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
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,
because_Cache,
setElimination,
rename,
independent_pairFormation,
Error :inhabitedIsType,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
Error :functionIsType,
equalityElimination,
productElimination,
promote_hyp,
instantiate,
cumulativity,
Error :inlEquality_alt,
Error :productIsType,
closedConclusion,
addEquality,
int_eqEquality,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isectIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
Error :functionExtensionality_alt
Latex:
\mforall{}beta:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}n0:finite-nat-seq(). \mforall{}x,n:\mBbbN{}.
((\mneg{}((beta x) = 0))
{}\mRightarrow{} (\mforall{}y:\mBbbN{}x. ((beta y) = 0))
{}\mRightarrow{} (\muparrow{}isl(gamma-neighbourhood(beta;n0) ext-finite-nat-seq(n0**\mlambda{}k.x\^{}(1);0)\^{}(n)))
{}\mRightarrow{} ((gamma-neighbourhood(beta;n0) ext-finite-nat-seq(n0**\mlambda{}k.x\^{}(1);0)\^{}(n)) = (inl 1)))
Date html generated:
2019_06_20-PM-03_04_28
Last ObjectModification:
2018_12_06-PM-11_34_52
Theory : continuity
Home
Index