Nuprl Lemma : decidable-cantor-to-int
∀[R:ℤ ⟶ ℤ ⟶ ℙ]. ((∀x,y:ℤ. Dec(R[x;y])) ⇒ (∀F:(ℕ ⟶ 𝔹) ⟶ ℤ. Dec(∃f,g:ℕ ⟶ 𝔹. R[F f;F g])))
Proof
Definitions occuring in Statement :
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],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
nat: ℕ,
sq_type: SQType(T),
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
bfalse: ff,
so_apply: x[s1;s2],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
false: False,
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
cantor-to-int-uniform-continuity,
nat_wf,
equal_wf,
set-value-type,
le_wf,
int-value-type,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable-finite-cantor-to-int,
bool_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
int_seg_wf,
lelt_wf,
all_wf,
decidable_wf,
not_wf,
exists_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
ifthenelse_wf,
bfalse_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
less_than_wf,
int_seg_properties,
nat_properties,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
rename,
cutEval,
dependent_set_memberEquality,
isectElimination,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
lambdaEquality,
independent_isectElimination,
intEquality,
natural_numberEquality,
setElimination,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
applyEquality,
functionExtensionality,
functionEquality,
because_Cache,
unionElimination,
equalityElimination,
independent_pairFormation,
universeEquality,
inlFormation,
inrFormation,
dependent_pairFormation,
voidElimination,
addLevel,
hyp_replacement,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
levelHypothesis
Latex:
\mforall{}[R:\mBbbZ{} {}\mrightarrow{} \mBbbZ{} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x,y:\mBbbZ{}. Dec(R[x;y])) {}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbZ{}. Dec(\mexists{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbB{}. R[F f;F g])))
Date html generated:
2017_04_17-AM-09_59_41
Last ObjectModification:
2017_02_27-PM-05_52_36
Theory : continuity
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