Nuprl Lemma : decidable-finite-cantor
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ((∀x,y:T. Dec(R[x;y])) ⇒ (∀n:ℕ. ∀F:(ℕn ⟶ 𝔹) ⟶ T. Dec(∃f,g:ℕn ⟶ 𝔹. R[F f;F g])))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
uimplies: b supposing a,
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
guard: {T},
subtype_rel: A ⊆r B,
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
le: A ≤ B,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
select: L[n],
cons: [a / b],
nil: [],
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
int_seg_wf,
bool_wf,
nat_wf,
all_wf,
decidable_wf,
equal_wf,
length_wf,
subtract_wf,
list_wf,
isect_wf,
sq_exists_wf,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
select_wf,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_rel_self,
set_wf,
less_than_wf,
primrec-wf2,
not_wf,
sq_stable_from_decidable,
squash_wf,
true_wf,
iff_weakening_equal,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
non_neg_length,
int_seg_subtype,
false_wf,
append_wf,
cons_wf,
btrue_wf,
nil_wf,
bfalse_wf,
length-append,
length-singleton,
add_functionality_wrt_eq,
bool_subtype_base,
select_append_front,
select-append,
subtype_rel_list,
top_wf,
int_seg_subtype_nat,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
not_functionality_wrt_uiff,
assert_wf,
select-cons-hd,
bool_cases,
length_of_cons_lemma,
length_of_nil_lemma,
exists_wf,
stuck-spread,
base_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
functionEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
cumulativity,
universeEquality,
axiomEquality,
intEquality,
because_Cache,
productEquality,
productElimination,
functionExtensionality,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
inlFormation,
inrFormation,
dependent_set_memberFormation,
independent_pairEquality,
hyp_replacement,
imageElimination,
imageMemberEquality,
baseClosed,
addEquality,
promote_hyp,
applyLambdaEquality,
equalityElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. Dec(R[x;y])) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}F:(\mBbbN{}n {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} T. Dec(\mexists{}f,g:\mBbbN{}n {}\mrightarrow{} \mBbbB{}. R[F f;F g])))
Date html generated:
2019_06_20-PM-02_49_45
Last ObjectModification:
2018_09_26-AM-09_54_23
Theory : continuity
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