Nuprl Lemma : weak-continuity-equipollent
∀T:Type. (T ~ ℕ ⇒ (∀F:(ℕ ⟶ T) ⟶ ℕ. ∀f:ℕ ⟶ T. ⇃(∃n:ℕ. ∀g:ℕ ⟶ T. ((f = g ∈ (ℕn ⟶ T)) ⇒ ((F f) = (F g) ∈ ℕ)))))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
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,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
equipollent: A ~ B,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
nat: ℕ,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
compose: f o g,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
biject-inverse,
nat_wf,
strong-continuity2-implies-weak,
compose_wf,
equipollent_wf,
exists_wf,
all_wf,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
implies-quotient-true,
int_seg_properties,
nat_properties,
decidable__le,
le_wf,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
and_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
cumulativity,
sqequalRule,
universeEquality,
because_Cache,
natural_numberEquality,
setElimination,
independent_isectElimination,
independent_pairFormation,
dependent_pairFormation,
dependent_set_memberEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
hyp_replacement,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}T:Type. (T \msim{} \mBbbN{} {}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} T) {}\mrightarrow{} \mBbbN{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} T. \00D9(\mexists{}n:\mBbbN{}. \mforall{}g:\mBbbN{} {}\mrightarrow{} T. ((f = g) {}\mRightarrow{} ((F f) = (F g))))))
Date html generated:
2017_09_29-PM-06_05_56
Last ObjectModification:
2017_07_05-PM-05_53_35
Theory : continuity
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