Nuprl Lemma : baire-diff-from-diff
∀a:ℕ ⟶ ℕ. ∀n:ℕ. (¬((a n) = (baire-diff-from(a;n) n) ∈ ℕ))
Proof
Definitions occuring in Statement :
baire-diff-from: baire-diff-from(a;k),
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
int_upper: {i...},
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
squash: ↓T,
sq_type: SQType(T),
bfalse: ff,
or: P ∨ Q,
decidable: Dec(P),
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat-pred: n-1,
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
baire-diff-from: baire-diff-from(a;k),
all: ∀x:A. B[x]
Lemmas referenced :
add_nat_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
int_upper_properties,
subtract_wf,
zero-add,
nequal-le-implies,
false_wf,
int_upper_subtype_nat,
not_assert_elim,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
btrue_neq_bfalse,
assert_elim,
bnot_wf,
bfalse_wf,
and_wf,
iff_weakening_equal,
btrue_wf,
eq_int_eq_true,
int_subtype_base,
subtype_base_sq,
eqff_to_assert,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
intformnot_wf,
intformle_wf,
decidable__le,
equal_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
nat_wf,
le_wf,
assert_of_eq_int,
eq_int_wf,
assert_of_le_int,
eqtt_to_assert,
bool_wf,
le_int_wf
Rules used in proof :
functionEquality,
hypothesis_subsumption,
int_eqReduceFalseSq,
promote_hyp,
universeEquality,
baseClosed,
imageMemberEquality,
imageElimination,
independent_functionElimination,
cumulativity,
instantiate,
addEquality,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
applyLambdaEquality,
functionExtensionality,
independent_pairFormation,
dependent_set_memberEquality,
applyEquality,
int_eqReduceTrueSq,
natural_numberEquality,
independent_isectElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
because_Cache,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalRule,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}n:\mBbbN{}. (\mneg{}((a n) = (baire-diff-from(a;n) n)))
Date html generated:
2017_04_21-AM-11_24_10
Last ObjectModification:
2017_04_20-PM-06_29_52
Theory : continuity
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