Nuprl Lemma : bag_all-cons
∀[T:Type]. ∀[x:T]. ∀[b:bag(T)]. ∀[f:T ⟶ 𝔹]. (bag_all(x.b;f) ~ (f x) ∧b bag_all(b;f))
Proof
Definitions occuring in Statement :
bag_all: bag_all(b;f),
cons-bag: x.b,
bag: bag(T),
band: p ∧b q,
bool: 𝔹,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
exists: ∃x:A. B[x],
bag_all: bag_all(b;f),
cons-bag: x.b,
bag-accum: bag-accum(v,x.f[v; x];init;bs),
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
band: p ∧b q,
ifthenelse: if b then t else f fi ,
btrue: tt,
implies: P ⇒ Q,
sq_type: SQType(T),
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q
Lemmas referenced :
subtype_base_sq,
bool_wf,
bool_subtype_base,
bag_to_squash_list,
list_accum_cons_lemma,
btrue_wf,
band-btrue,
list_induction,
all_wf,
equal_wf,
list_accum_wf,
eqtt_to_assert,
list_wf,
list_accum_nil_lemma,
band_assoc,
band_wf,
bag_all_wf,
cons-bag_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
because_Cache,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
rename,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
applyEquality,
functionExtensionality,
lambdaEquality,
unionElimination,
equalityElimination,
hyp_replacement,
applyLambdaEquality,
sqequalAxiom,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[b:bag(T)]. \mforall{}[f:T {}\mrightarrow{} \mBbbB{}]. (bag\_all(x.b;f) \msim{} (f x) \mwedge{}\msubb{} bag\_all(b;f))
Date html generated:
2017_10_01-AM-08_52_23
Last ObjectModification:
2017_07_26-PM-04_33_58
Theory : bags
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