Nuprl Lemma : l_all-cons
∀[T:Type]. ∀x:T. ∀L:T List. ∀[P:{a:T| (a ∈ [x / L])} ⟶ ℙ]. ((∀a∈[x / L].P[a]) ⇐⇒ P[x] ∧ (∀a∈L.P[a]))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
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],
member: t ∈ T,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
l_all_cons,
l_member_wf,
cons_wf,
cons_member,
list-subtype,
subtype_rel_list_set,
equal_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
inlFormation,
dependent_set_memberEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
applyEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
setElimination,
rename,
inrFormation,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}x:T. \mforall{}L:T List. \mforall{}[P:\{a:T| (a \mmember{} [x / L])\} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}a\mmember{}[x / L].P[a]) \mLeftarrow{}{}\mRightarrow{} P[x] \mwedge{} (\mforall{}a\mmember{}L.P[a]))
Date html generated:
2016_05_14-AM-07_49_48
Last ObjectModification:
2015_12_26-PM-04_45_40
Theory : list_1
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