Nuprl Lemma : uniform-continuity-pi-dec
∀T:Type. ∀F:(ℕ ⟶ 𝔹) ⟶ T. ∀n:ℕ. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ ucA(T;F;n) ⇒ (∀m:ℕ. (m < n ⇒ Dec(ucA(T;F;m)))))
Proof
Definitions occuring in Statement :
uniform-continuity-pi: ucA(T;F;n),
nat: ℕ,
bool: 𝔹,
less_than: a < b,
decidable: Dec(P),
all: ∀x:A. B[x],
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,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
prop: ℙ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uniform-continuity-pi2: ucB(T;F;n),
uniform-continuity-pi: ucA(T;F;n),
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
sq_type: SQType(T),
guard: {T},
lelt: i ≤ j < k,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
ext2Cantor: ext2Cantor(n;f;d),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
so_lambda: λ2x.t[x],
so_apply: x[s],
bnot: ¬bb,
subtract: n - m,
label: ...$L... t,
squash: ↓T
Lemmas referenced :
istype-less_than,
uniform-continuity-pi_wf,
decidable_wf,
equal_wf,
istype-nat,
bool_wf,
istype-universe,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
uniform-continuity-pi2_wf,
int_seg_wf,
ext2Cantor_wf,
btrue_wf,
bfalse_wf,
eq_ext2Cantor,
subtype_rel_function,
nat_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
decidable__equal_bool,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
decidable__lt,
intformless_wf,
intformeq_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
iff_imp_equal_bool,
assert_elim,
bool_subtype_base,
istype-assert,
bool_cases,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
set_subtype_base,
le_wf,
lelt_wf,
bool_cases_sqequal,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
int_seg_properties,
decidable_functionality,
uniform-continuity-pi2-dec-ext,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
primrec-wf2,
all_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
squash_wf,
true_wf,
iff_weakening_equal,
int_seg_subtype,
not-le-2,
condition-implies-le,
minus-one-mul,
minus-add,
minus-minus,
minus-one-mul-top,
less-iff-le,
add_functionality_wrt_le,
le-add-cancel,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
Error :inhabitedIsType,
Error :universeIsType,
sqequalRule,
Error :functionIsType,
because_Cache,
instantiate,
universeEquality,
Error :dependent_set_memberEquality_alt,
addEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :equalityIsType1,
applyEquality,
functionExtensionality,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
Error :productIsType,
applyLambdaEquality,
equalityElimination,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
promote_hyp,
Error :setIsType,
functionEquality,
Error :inlFormation_alt,
imageElimination,
imageMemberEquality,
Error :inrFormation_alt,
minusEquality
Latex:
\mforall{}T:Type. \mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} T. \mforall{}n:\mBbbN{}.
((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} ucA(T;F;n) {}\mRightarrow{} (\mforall{}m:\mBbbN{}. (m < n {}\mRightarrow{} Dec(ucA(T;F;m)))))
Date html generated:
2019_06_20-PM-02_53_17
Last ObjectModification:
2018_10_30-PM-02_45_33
Theory : continuity
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