Nuprl Lemma : markov-streamless-iff-not-not-enum
(∀P:ℕ ⟶ ℙ. ((∀m:ℕ. ((P m) ∨ (¬(P m)))) ⇒ (¬(∀m:ℕ. (¬(P m)))) ⇒ (∃m:ℕ. (P m))))
⇒ (∀T:Type. (streamless(T) ⇐⇒ (∀x,y:T. Dec(x = y ∈ T)) ∧ (¬¬(∃L:T List. ∀x:T. (x ∈ L)))))
Proof
Definitions occuring in Statement :
streamless: streamless(T),
l_member: (x ∈ l),
list: T List,
nat: ℕ,
decidable: Dec(P),
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
not: ¬A,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
streamless: streamless(T),
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
l_member: (x ∈ l),
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than: a < b,
squash: ↓T,
pi1: fst(t),
inject: Inj(A;B;f),
true: True,
uiff: uiff(P;Q)
Lemmas referenced :
streamless-dec-equal,
streamless-implies-not-not-enum,
not_wf,
exists_wf,
list_wf,
all_wf,
l_member_wf,
streamless_wf,
decidable_wf,
equal_wf,
nat_wf,
or_wf,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
decidable__exists_int_seg,
decidable__cand,
decidable__not,
decidable__equal_nat,
lelt_wf,
length_wf,
select_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
non_neg_length,
le_wf,
length_wf_nat,
squash_wf,
true_wf,
iff_weakening_equal,
less_than_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__equal_int,
pigeon-hole,
add_nat_wf,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
subtype_rel_dep_function,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_functionElimination,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
cumulativity,
sqequalRule,
lambdaEquality,
productElimination,
productEquality,
universeEquality,
instantiate,
functionEquality,
applyEquality,
functionExtensionality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
isect_memberEquality,
unionElimination,
inlFormation,
inrFormation,
dependent_pairFormation,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
imageElimination,
promote_hyp,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
addEquality,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
(\mforall{}P:\mBbbN{} {}\mrightarrow{} \mBbbP{}. ((\mforall{}m:\mBbbN{}. ((P m) \mvee{} (\mneg{}(P m)))) {}\mRightarrow{} (\mneg{}(\mforall{}m:\mBbbN{}. (\mneg{}(P m)))) {}\mRightarrow{} (\mexists{}m:\mBbbN{}. (P m))))
{}\mRightarrow{} (\mforall{}T:Type. (streamless(T) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x,y:T. Dec(x = y)) \mwedge{} (\mneg{}\mneg{}(\mexists{}L:T List. \mforall{}x:T. (x \mmember{} L)))))
Date html generated:
2018_05_21-PM-09_03_01
Last ObjectModification:
2017_07_26-PM-06_25_50
Theory : general
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