Nuprl Lemma : assert-bl-all
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ 𝔹]. uiff(↑(∀x∈L.P[x])_b;(∀x∈L.↑P[x]))
Proof
Definitions occuring in Statement :
bl-all: (∀x∈L.P[x])_b,
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
bl-all: (∀x∈L.P[x])_b,
all: ∀x:A. B[x],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
band: p ∧b q,
bfalse: ff,
less_than: a < b,
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
list_induction,
uiff_wf,
assert_wf,
bl-all_wf,
l_member_wf,
l_all_wf,
list_wf,
reduce_nil_lemma,
l_all_nil,
assert_witness,
select_wf,
nil_wf,
length_of_nil_lemma,
int_seg_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,
int_seg_wf,
length_wf,
true_wf,
l_all_wf_nil,
reduce_cons_lemma,
l_all_cons,
bool_cases_sqequal,
cons_wf,
length_of_cons_lemma,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
band_wf,
assert_of_band,
bool_wf,
assert_elim,
and_wf,
equal_wf,
list-subtype
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
applyEquality,
functionExtensionality,
setElimination,
rename,
hypothesis,
setEquality,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
independent_isectElimination,
natural_numberEquality,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
addEquality,
pointwiseFunctionality,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
independent_pairEquality,
functionEquality,
universeEquality,
hyp_replacement,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}]. uiff(\muparrow{}(\mforall{}x\mmember{}L.P[x])\_b;(\mforall{}x\mmember{}L.\muparrow{}P[x]))
Date html generated:
2016_10_21-AM-10_15_27
Last ObjectModification:
2016_07_12-AM-05_31_45
Theory : list_1
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