Nuprl Lemma : bnot_thru_exists
∀A:Type. ∀as:A List. ∀f:A ⟶ 𝔹. ¬b(∃bx(:A) ∈ as. f[x]) = ∀bx(:A) ∈ as. (¬bf[x])
Proof
Definitions occuring in Statement :
bexists: bexists,
ball: ball,
list: T List,
bnot: ¬bb,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
squash: ↓T,
prop: ℙ,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
false: False,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
all_wf,
bool_wf,
equal_wf,
bnot_wf,
bexists_wf,
ball_wf,
list_wf,
bexists_nil_lemma,
ball_nil_lemma,
btrue_wf,
bexists_cons_lemma,
ball_cons_lemma,
bnot_thru_bor,
squash_wf,
true_wf,
band_wf,
eqtt_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
eqff_to_assert,
assert-bnot,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
hypothesis,
dependent_functionElimination,
applyEquality,
functionExtensionality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
imageElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
universeEquality
Latex:
\mforall{}A:Type. \mforall{}as:A List. \mforall{}f:A {}\mrightarrow{} \mBbbB{}. \mneg{}\msubb{}(\mexists{}\msubb{}x(:A) \mmember{} as. f[x]) = \mforall{}\msubb{}x(:A) \mmember{} as. (\mneg{}\msubb{}f[x])
Date html generated:
2017_10_01-AM-09_56_00
Last ObjectModification:
2017_03_03-PM-00_56_08
Theory : list_2
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