Nuprl Lemma : sq_stable__l_all
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ ℙ]. ((∀[x:{x:T| (x ∈ L)} ]. SqStable(P[x])) ⇒ SqStable((∀x∈L.P[x])))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
l_all: (∀x∈L.P[x]),
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
and: P ∧ Q,
squash: ↓T,
all: ∀x:A. B[x]
Lemmas referenced :
sq_stable__all,
sq_stable__le,
list-subtype,
select_wf,
length_wf,
int_seg_wf,
list_wf,
sq_stable_wf,
l_member_wf,
uall_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
natural_numberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination,
introduction,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}].
((\mforall{}[x:\{x:T| (x \mmember{} L)\} ]. SqStable(P[x])) {}\mRightarrow{} SqStable((\mforall{}x\mmember{}L.P[x])))
Date html generated:
2016_05_14-AM-06_40_36
Last ObjectModification:
2016_01_14-PM-08_19_58
Theory : list_0
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