Nuprl Lemma : ball_functionality_wrt_bimplies
∀T:Type. ∀as:T List. ∀P,Q:T ⟶ 𝔹. ((∀x:T. (↑(P[x] ⇒b Q[x]))) ⇒ (↑(∀bx(:T) ∈ as. P[x] ⇒b (∀bx(:T) ∈ as. Q[x]))))
Proof
Definitions occuring in Statement :
ball: ball,
list: T List,
bimplies: p ⇒b q,
assert: ↑b,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
bimplies: p ⇒b q,
top: Top,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
btrue: tt,
bor: p ∨bq,
bfalse: ff,
assert: ↑b,
true: True,
ball: ball,
or: P ∨ Q,
guard: {T},
rev_implies: P ⇐ Q,
decidable: Dec(P),
cand: A c∧ B,
sq_type: SQType(T)
Lemmas referenced :
iff_weakening_uiff,
assert_wf,
bimplies_wf,
isect_wf,
assert_of_bimplies,
list_induction,
bor_wf,
bnot_wf,
ball_wf,
list_wf,
ball_nil_lemma,
ball_cons_lemma,
bool_wf,
equal_wf,
iff_transitivity,
band_wf,
or_wf,
not_wf,
assert_of_bor,
uiff_transitivity,
assert_of_bnot,
not_functionality_wrt_uiff,
assert_of_band,
decidable__and2,
decidable__assert,
assert_elim,
and_wf,
bfalse_wf,
subtype_base_sq,
bool_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
cumulativity,
because_Cache,
applyLambdaEquality,
isect_memberEquality,
sqequalRule,
lambdaEquality,
independent_functionElimination,
productElimination,
voidElimination,
voidEquality,
natural_numberEquality,
rename,
functionIsType,
universeIsType,
inhabitedIsType,
universeEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
independent_pairFormation,
unionElimination,
inlFormation,
inrFormation,
independent_isectElimination,
dependent_set_memberEquality,
setElimination,
instantiate
Latex:
\mforall{}T:Type. \mforall{}as:T List. \mforall{}P,Q:T {}\mrightarrow{} \mBbbB{}.
((\mforall{}x:T. (\muparrow{}(P[x] {}\mRightarrow{}\msubb{} Q[x]))) {}\mRightarrow{} (\muparrow{}(\mforall{}\msubb{}x(:T) \mmember{} as. P[x] {}\mRightarrow{}\msubb{} (\mforall{}\msubb{}x(:T) \mmember{} as. Q[x]))))
Date html generated:
2019_10_16-PM-01_05_22
Last ObjectModification:
2018_09_26-PM-08_33_02
Theory : list_2
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