Nuprl Lemma : uniform-continuity-pi-pi-prop2
∀T:Type. ∀F:(ℕ ⟶ 𝔹) ⟶ T. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (∃n:ℕ. ucpB(T;F;n) ⇐⇒ ∃n:ℕ. ucA(T;F;n)))
Proof
Definitions occuring in Statement :
uniform-continuity-pi-pi: ucpB(T;F;n),
uniform-continuity-pi: ucA(T;F;n),
nat: ℕ,
bool: 𝔹,
decidable: Dec(P),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
uniform-continuity-pi-pi: ucpB(T;F;n),
uimplies: b supposing a,
nat: ℕ,
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
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,
uniform-continuity-pi: ucA(T;F;n),
sq_stable: SqStable(P)
Lemmas referenced :
exists_wf,
nat_wf,
uniform-continuity-pi-pi_wf,
bool_wf,
uniform-continuity-pi_wf,
all_wf,
decidable_wf,
equal_wf,
set-value-type,
le_wf,
int-value-type,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
uniform-continuity-pi-dec,
int_seg_subtype_nat,
false_wf,
int_seg_properties,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
int_seg_wf,
uniform-continuity-pi-search_wf,
subtype_rel_union,
not_wf,
top_wf,
or_wf,
uniform-continuity-pi-search-prop2,
itermAdd_wf,
itermConstant_wf,
int_term_value_add_lemma,
int_term_value_constant_lemma,
lelt_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
subtype_rel_dep_function,
subtype_rel_self,
squash_wf,
sq_stable__all,
sq_stable__equal,
sq_stable__le,
le_weakening2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
functionEquality,
universeEquality,
productElimination,
dependent_pairFormation,
cutEval,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
intEquality,
natural_numberEquality,
setElimination,
rename,
instantiate,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache,
addEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
axiomEquality
Latex:
\mforall{}T:Type. \mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} T. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mexists{}n:\mBbbN{}. ucpB(T;F;n) \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}. ucA(T;F;n)))
Date html generated:
2017_04_17-AM-09_59_09
Last ObjectModification:
2017_02_27-PM-05_53_20
Theory : continuity
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