Nuprl Lemma : gamma-neighbourhood-prop1
∀beta:ℕ ⟶ ℕ. ∀n0:finite-nat-seq().
((∀a:ℕ ⟶ ℕ. ∃x:ℕ. (↑isl(gamma-neighbourhood(beta;n0) a^(x))))
∧ (∀a,b:finite-nat-seq().
((↑isl(gamma-neighbourhood(beta;n0) a))
⇒ ((gamma-neighbourhood(beta;n0) a) = (gamma-neighbourhood(beta;n0) a**b) ∈ (ℕ?)))))
Proof
Definitions occuring in Statement :
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],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
isl: isl(x),
exists: ∃x:A. B[x],
nat: ℕ,
finite-nat-seq: finite-nat-seq(),
pi1: fst(t),
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
gamma-neighbourhood: gamma-neighbourhood(beta;n0),
ifthenelse: if b then t else f fi ,
btrue: tt,
assert: ↑b,
bfalse: ff,
init-seg-nat-seq: init-seg-nat-seq(f;g),
iff: P ⇐⇒ Q,
mk-finite-nat-seq: f^(n),
append-finite-nat-seq: f**g,
pi2: snd(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
sq_type: SQType(T),
bnot: ¬bb,
squash: ↓T,
rev_implies: P ⇐ Q
Lemmas referenced :
istype-assert,
gamma-neighbourhood_wf,
btrue_wf,
bfalse_wf,
finite-nat-seq_wf,
istype-nat,
add_nat_wf,
istype-false,
istype-le,
nat_properties,
decidable__le,
add-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
mk-finite-nat-seq_wf,
subtype_rel_function,
nat_wf,
int_seg_wf,
int_seg_subtype_nat,
subtype_rel_self,
bool_cases_sqequal,
init-seg-nat-seq_wf,
assert_of_tt,
assert-init-seg-nat-seq,
extend-seq1-all-dec,
all_wf,
decidable_wf,
exists_wf,
assert_wf,
append-finite-nat-seq_wf,
not_wf,
equal-wf-base,
set_subtype_base,
le_wf,
int_subtype_base,
eqtt_to_assert,
eqff_to_assert,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
init-seg-nat-seq-append-implies-left,
decidable__equal_int,
unit_wf2,
true_wf,
init-seg-nat-seq-append-implies-right,
decidable__exists_int_seg,
decidable__not,
equal_wf,
squash_wf,
istype-universe,
iff_weakening_equal,
append-finite-nat-seq-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
Error :inhabitedIsType,
hypothesisEquality,
independent_pairFormation,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
unionElimination,
sqequalRule,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
Error :universeIsType,
Error :functionIsType,
Error :dependent_pairFormation_alt,
Error :dependent_set_memberEquality_alt,
addEquality,
productElimination,
setElimination,
rename,
closedConclusion,
natural_numberEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
baseClosed,
independent_isectElimination,
approximateComputation,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
because_Cache,
instantiate,
functionEquality,
productEquality,
intEquality,
equalityElimination,
cumulativity,
Error :inlEquality_alt,
Error :productIsType,
sqequalBase,
Error :equalityIsType1,
Error :inrFormation_alt,
Error :inlFormation_alt,
Error :unionIsType,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}beta:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}n0:finite-nat-seq().
((\mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mexists{}x:\mBbbN{}. (\muparrow{}isl(gamma-neighbourhood(beta;n0) a\^{}(x))))
\mwedge{} (\mforall{}a,b:finite-nat-seq().
((\muparrow{}isl(gamma-neighbourhood(beta;n0) a))
{}\mRightarrow{} ((gamma-neighbourhood(beta;n0) a) = (gamma-neighbourhood(beta;n0) a**b)))))
Date html generated:
2019_06_20-PM-03_03_53
Last ObjectModification:
2018_11_28-AM-08_57_16
Theory : continuity
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