Nuprl Lemma : list_all_iff
∀[T:Type]. ∀l:T List. ∀[P:T ⟶ ℙ]. (list_all(x.P[x];l) ⇐⇒ ∀x:T. ((x ∈ l) ⇒ P[x]))
Proof
Definitions occuring in Statement :
list_all: list_all(x.P[x];l),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
list_all: list_all(x.P[x];l),
top: Top,
uimplies: b supposing a,
not: ¬A,
false: False,
true: True,
or: P ∨ Q,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
list_induction,
uall_wf,
iff_wf,
list_all_wf,
all_wf,
l_member_wf,
list_wf,
reduce_nil_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
true_wf,
reduce_cons_lemma,
cons_member,
and_wf,
equal_wf,
cons_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
universeEquality,
applyEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
functionIsType,
universeIsType,
rename,
productElimination,
unionElimination,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
productEquality,
inlFormation,
inrFormation,
inhabitedIsType
Latex:
\mforall{}[T:Type]. \mforall{}l:T List. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. (list\_all(x.P[x];l) \mLeftarrow{}{}\mRightarrow{} \mforall{}x:T. ((x \mmember{} l) {}\mRightarrow{} P[x]))
Date html generated:
2019_10_15-AM-10_54_00
Last ObjectModification:
2018_09_27-AM-10_02_43
Theory : list!
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