Nuprl Lemma : gamma-neighbourhood-prop2
∀beta:ℕ ⟶ ℕ. ∀n0:finite-nat-seq(). ∀x:ℕ.
((¬((beta x) = 0 ∈ ℤ))
⇒ (∃x1:ℕx + 1
((¬((beta x1) = 0 ∈ ℤ))
∧ (∀y:ℕx1. ((beta y) = 0 ∈ ℤ))
∧ ((gamma-neighbourhood(beta;n0) n0**λx.x1^(1)) = (inl 1) ∈ (ℕ?))
∧ ((gamma-neighbourhood(beta;n0) n0**λx.(x1 + 1)^(1)) = (inl 0) ∈ (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(),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: 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,
not: ¬A,
member: t ∈ T,
subtype_rel: A ⊆r B,
nat: ℕ,
false: False,
so_apply: x[s],
so_lambda: λ2x.t[x],
less_than': less_than'(a;b),
le: A ≤ B,
prop: ℙ,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
exists: ∃x:A. B[x],
sq_type: SQType(T),
cand: A c∧ B,
true: True,
squash: ↓T,
less_than: a < b,
guard: {T},
ge: i ≥ j ,
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 ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
pi2: snd(t),
pi1: fst(t),
append-finite-nat-seq: f**g,
mk-finite-nat-seq: f^(n),
finite-nat-seq: finite-nat-seq(),
rev_implies: P ⇐ Q
Lemmas referenced :
istype-int,
istype-nat,
istype-void,
finite-nat-seq_wf,
int_subtype_base,
le_wf,
set_subtype_base,
istype-false,
int_seg_subtype_nat,
istype-less_than,
istype-le,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
full-omega-unsat,
decidable__lt,
all_wf,
equal-wf-base,
not_wf,
exists_wf,
primrec-wf2,
subtract_wf,
int_seg_wf,
subtype_base_sq,
int_formula_prop_le_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformle_wf,
intformeq_wf,
intformand_wf,
decidable__equal_int,
int_seg_properties,
decidable__not,
dec-exists-int-seg,
subtract-add-cancel,
unit_wf2,
gamma-neighbourhood_wf,
append-finite-nat-seq_wf,
mk-finite-nat-seq_wf,
decidable__le,
union_subtype_base,
nat_wf,
unit_subtype_base,
nat_properties,
init-seg-nat-seq_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
assert-init-seg-nat-seq2,
extend-seq1-all-dec,
subtype_rel_self,
decidable_wf,
assert_wf,
istype-top,
less_than_anti-reflexive,
less_than_wf,
lelt_wf,
istype-assert,
assert-init-seg-nat-seq,
equal_wf,
squash_wf,
true_wf,
istype-universe,
append-empty-nat-seq,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
sqequalRule,
Error :functionIsType,
Error :equalityIstype,
cut,
introduction,
extract_by_obid,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
Error :lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
rename,
natural_numberEquality,
Error :universeIsType,
because_Cache,
Error :inhabitedIsType,
equalitySymmetry,
sqequalBase,
baseClosed,
closedConclusion,
intEquality,
Error :productIsType,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
isectElimination,
unionElimination,
addEquality,
dependent_functionElimination,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
Error :dependent_pairFormation_alt,
productEquality,
functionEquality,
instantiate,
Error :setIsType,
cumulativity,
imageMemberEquality,
equalityTransitivity,
productElimination,
Error :unionIsType,
equalityElimination,
promote_hyp,
Error :inlEquality_alt,
hyp_replacement,
applyLambdaEquality,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isectIsTypeImplies,
imageElimination,
baseApply,
universeEquality
Latex:
\mforall{}beta:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}n0:finite-nat-seq(). \mforall{}x:\mBbbN{}.
((\mneg{}((beta x) = 0))
{}\mRightarrow{} (\mexists{}x1:\mBbbN{}x + 1
((\mneg{}((beta x1) = 0))
\mwedge{} (\mforall{}y:\mBbbN{}x1. ((beta y) = 0))
\mwedge{} ((gamma-neighbourhood(beta;n0) n0**\mlambda{}x.x1\^{}(1)) = (inl 1))
\mwedge{} ((gamma-neighbourhood(beta;n0) n0**\mlambda{}x.(x1 + 1)\^{}(1)) = (inl 0)))))
Date html generated:
2019_06_20-PM-03_04_05
Last ObjectModification:
2018_12_06-PM-11_58_06
Theory : continuity
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