Nuprl Lemma : decidable__l_exists-proof
∀[A:Type]. ∀[F:A ⟶ ℙ]. ∀L:A List. ((∀k:A. Dec(F[k])) ⇒ Dec((∃k∈L. F[k])))
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
l_exists: (∃x∈L. P[x]),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T
Lemmas referenced :
list_wf,
decidable_wf,
all_wf,
int_seg_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
select_wf,
length_wf,
decidable__exists_int_seg
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
natural_numberEquality,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
cumulativity,
setElimination,
rename,
independent_isectElimination,
because_Cache,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
introduction,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[F:A {}\mrightarrow{} \mBbbP{}]. \mforall{}L:A List. ((\mforall{}k:A. Dec(F[k])) {}\mRightarrow{} Dec((\mexists{}k\mmember{}L. F[k])))
Date html generated:
2016_05_14-AM-07_47_52
Last ObjectModification:
2016_01_15-AM-08_33_00
Theory : list_1
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