Nuprl Lemma : not-assert-bl-all
∀[T:Type]. ∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ 𝔹. (¬↑(∀x∈L.P[x])_b ⇐⇒ (∃x∈L. ¬↑P[x]))
Proof
Definitions occuring in Statement :
bl-all: (∀x∈L.P[x])_b,
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
l_all: (∀x∈L.P[x]),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than: a < b,
squash: ↓T,
l_exists: (∃x∈L. P[x])
Lemmas referenced :
length_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,
assert-bl-all,
set_wf,
decidable__assert,
decidable__not,
list-subtype,
decidable__l_exists,
list_wf,
bool_wf,
l_exists_wf,
l_member_wf,
bl-all_wf,
assert_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
hypothesis,
independent_functionElimination,
voidElimination,
functionEquality,
universeEquality,
dependent_functionElimination,
cumulativity,
equalityTransitivity,
equalitySymmetry,
unionElimination,
productElimination,
independent_isectElimination,
because_Cache,
setElimination,
rename,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}. (\mneg{}\muparrow{}(\mforall{}x\mmember{}L.P[x])\_b \mLeftarrow{}{}\mRightarrow{} (\mexists{}x\mmember{}L. \mneg{}\muparrow{}P[x]))
Date html generated:
2016_05_14-PM-02_10_53
Last ObjectModification:
2016_01_15-AM-08_00_23
Theory : list_1
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