Nuprl Lemma : gamma-neighbourhood-prop3
∀beta:ℕ ⟶ ℕ. ∀n,m:ℕ.
(((beta 0) = 0 ∈ ℤ)
⇒ (↑isl(gamma-neighbourhood(beta;0s^(n)) 0s^(m)))
⇒ (n < m ∧ ((gamma-neighbourhood(beta;0s^(n)) 0s^(m)) = (inl 0) ∈ (ℕ?))))
Proof
Definitions occuring in Statement :
gamma-neighbourhood: gamma-neighbourhood(beta;n0),
mk-finite-nat-seq: f^(n),
zero-seq: 0s,
nat: ℕ,
assert: ↑b,
isl: isl(x),
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
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,
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
isl: isl(x),
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
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,
ge: i ≥ j ,
iff: P ⇐⇒ Q,
zero-seq: 0s,
mk-finite-nat-seq: f^(n),
append-finite-nat-seq: f**g,
pi1: fst(t),
pi2: snd(t),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
true: True,
squash: ↓T,
cand: A c∧ B,
rev_implies: P ⇐ Q,
finite-nat-seq: finite-nat-seq()
Lemmas referenced :
istype-assert,
gamma-neighbourhood_wf,
mk-finite-nat-seq_wf,
zero-seq_wf,
subtype_rel_function,
nat_wf,
int_seg_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
btrue_wf,
bfalse_wf,
istype-int,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
set_subtype_base,
le_wf,
int_subtype_base,
istype-nat,
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,
finite-nat-seq_wf,
decidable_wf,
assert_wf,
append-finite-nat-seq_wf,
not_wf,
equal-wf-base,
true_wf,
nat_properties,
intformand_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
assert-init-seg-nat-seq2,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
istype-less_than,
istype-top,
less_than_anti-reflexive,
less_than_wf,
decidable__equal_int,
unit_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
hypothesis,
because_Cache,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
Error :inhabitedIsType,
unionElimination,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
Error :dependent_set_memberEquality_alt,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
voidElimination,
Error :universeIsType,
intEquality,
baseClosed,
sqequalBase,
Error :functionIsType,
equalityElimination,
productElimination,
promote_hyp,
instantiate,
cumulativity,
functionEquality,
productEquality,
int_eqEquality,
hyp_replacement,
applyLambdaEquality,
addEquality,
Error :productIsType,
baseApply,
closedConclusion,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isectIsTypeImplies,
imageMemberEquality,
imageElimination,
Error :inlEquality_alt,
Error :functionExtensionality_alt
Latex:
\mforall{}beta:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}n,m:\mBbbN{}.
(((beta 0) = 0)
{}\mRightarrow{} (\muparrow{}isl(gamma-neighbourhood(beta;0s\^{}(n)) 0s\^{}(m)))
{}\mRightarrow{} (n < m \mwedge{} ((gamma-neighbourhood(beta;0s\^{}(n)) 0s\^{}(m)) = (inl 0))))
Date html generated:
2019_06_20-PM-03_04_12
Last ObjectModification:
2018_12_06-PM-11_34_57
Theory : continuity
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