Nuprl Lemma : decidable__l_all-proof
∀[A:Type]. ∀L:A List. ∀[F:{a:A| (a ∈ L)} ⟶ ℙ]. ((∀k:{a:A| (a ∈ L)} . Dec(F[k])) ⇒ Dec((∀k∈L.F[k])))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
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,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
l_all: (∀x∈L.P[x]),
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
guard: {T},
int_seg: {i..j-},
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,
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,
list-subtype,
l_member_wf,
select_wf,
length_wf,
decidable__all_int_seg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
natural_numberEquality,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
setElimination,
rename,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
cumulativity,
independent_functionElimination,
introduction,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}L:A List. \mforall{}[F:\{a:A| (a \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}k:\{a:A| (a \mmember{} L)\} . Dec(F[k])) {}\mRightarrow{} Dec((\mforall{}k\mmember{}L.F[k])\000C))
Date html generated:
2016_05_14-AM-07_47_33
Last ObjectModification:
2016_01_15-AM-08_33_32
Theory : list_1
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