Nuprl Lemma : weak-continuity-nat-int-bool
∀F:(ℕ ⟶ ℤ) ⟶ 𝔹. ∀f:ℕ ⟶ ℤ. ⇃(∃n:ℕ. ∀g:ℕ ⟶ ℤ. ((f = g ∈ (ℕn ⟶ ℤ)) ⇒ F f = F g))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
weak-continuity-nat-int,
nat_wf,
bool_wf,
eqtt_to_assert,
false_wf,
le_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
exists_wf,
all_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
subtype_rel_self,
implies-quotient-true,
btrue_wf,
equal-wf-base,
nat_properties,
full-omega-unsat,
intformeq_wf,
itermConstant_wf,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
bfalse_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
functionEquality,
intEquality,
because_Cache,
unionElimination,
equalityElimination,
sqequalRule,
isectElimination,
productElimination,
independent_isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
setElimination,
rename,
baseClosed,
applyLambdaEquality,
approximateComputation,
isect_memberEquality,
voidEquality
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbZ{}) {}\mrightarrow{} \mBbbB{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbZ{}. \00D9(\mexists{}n:\mBbbN{}. \mforall{}g:\mBbbN{} {}\mrightarrow{} \mBbbZ{}. ((f = g) {}\mRightarrow{} F f = F g))
Date html generated:
2017_09_29-PM-06_06_35
Last ObjectModification:
2017_07_05-PM-06_21_55
Theory : continuity
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