Nuprl Lemma : can-find-first
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀L:T List. ((∃x:T [first-member(T;x;L;P)]) ∨ (∀x∈L.¬↑(P x)))
Proof
Definitions occuring in Statement :
first-member: first-member(T;x;L;P),
l_all: (∀x∈L.P[x]),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
guard: {T},
or: P ∨ Q,
top: Top,
sq_exists: ∃x:A [B[x]],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
decidable: Dec(P),
cand: A c∧ B,
first-member: first-member(T;x;L;P),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
satisfiable_int_formula: satisfiable_int_formula(fmla),
select: L[n],
cons: [a / b]
Lemmas referenced :
list_induction,
or_wf,
sq_exists_wf,
first-member_wf,
l_all_wf2,
not_wf,
assert_wf,
l_member_wf,
list_wf,
l_all_nil,
nil_wf,
bool_wf,
cons_wf,
ifthenelse_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
first-member-cons,
decidable__assert,
l_all_cons,
length_of_cons_lemma,
false_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
lelt_wf,
length_wf,
int_seg_properties,
intformle_wf,
int_formula_prop_le_lemma,
select_wf,
int_seg_wf,
decidable__le,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
functionExtensionality,
applyEquality,
hypothesis,
setElimination,
rename,
setEquality,
independent_functionElimination,
inrFormation,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
dependent_functionElimination,
functionEquality,
universeEquality,
unionElimination,
inlFormation,
dependent_set_memberEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_pairFormation,
dependent_set_memberFormation,
natural_numberEquality,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
int_eqEquality,
intEquality,
computeAll,
addEquality,
productEquality,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List. ((\mexists{}x:T [first-member(T;x;L;P)]) \mvee{} (\mforall{}x\mmember{}L.\mneg{}\muparrow{}(P x)))
Date html generated:
2018_05_21-PM-06_34_04
Last ObjectModification:
2017_07_26-PM-04_52_22
Theory : general
Home
Index